2666
Znd. Eng. Chem. Res. 1995,34, 2566-2573
Termination of Computer-Generated Reaction Mechanisms: Species Rank-Based Convergence Criterion Linda J. Broadbelt,*vt Scott M. Stark, and Michael T. Klein*” Center for Catalytic Science and Technology, Department of Chemical Engineering, University of Delaware, Newark, Delaware 19716
The issue of the termination or truncation of computer-generated reaction models is addressed. The product rank of each species, a quantitative measure of the order in which a product appears in a reaction mechanism, was tallied during generation of the reaction mechanism. Incorporation of product rank in the convergence criterion allowed generation of reaction mechanisms for reactants and conditions that previously exhausted available memory when only the maximum carbon count was specified. A rank-based convergence criterion alone resulted in generation of infinite reaction mechanisms when radical addition reactions were considered. The rank-based criterion combined with a n upper bound on the atom count of radical species capped the number of addition reactions that occurred. Coupling these two criteria provided the best capability for predicting reactant conversion and major and minor product yields with the fewest number of excess components. Inherently convergent reaction networks require no external convergence criteria.
Introduction The intense activity in molecular reaction engineering has generated several approaches to the creation of molecular reaction models by computer (Chevalier et al., 1990; Clymans and Froment, 1984; DiMaio and Lignola, 1992; Hillewaert et al., 1988; Quann and Jaffe, 1992). The essential idea is to input some representation of reactant structure and chemical reaction. Algorithms and grammar for representing and determining species connectivity, uniqueness, and relationships (the reaction network) create an output that is, in some form, the controlling ordinary differential equations of the reaction model. One of the subtle challenges confronting the generalizability of such model builders is the divergent nature of certain key classes of chemistry. For example, in pyrolysis chemistry, where radical addition to olefins can be important, the mechanism can, in principle, grow without bound. Thus some external criterion for the rational halt of mechanism generation is required. In our previous use of alkane pyrolysis chemistry to illustrate a graph theory-based mechanism generator (Broadbelt et al., 19941, the simple carbon-count convergence criterion was used. The user-specified carboncount convergence criterion impacted the number of reactions included and therefore the number of species and reactions generated. Depending on the carbon count used, a wide range of outputs for ethane pyrolysis was found. Allowing reaction of only species with two or fewer carbon atoms generated 11 species and 55 equations. When this termination criterion was increased to three carbon atoms, 99 species and 611 equations were created. The number of species and equations as a function of the carbon-number stopping criterion increased essentially exponentially with increasing carbon number for carbon numbers up to 4. The species and equation numbers and even CPU demand were a strong function of the stopping criterion but insensitive to the starting alkane reactant. The Present address: 2145 Sheridan Road, Department of Chemical Engineering, Northwestern University, Evanston, IL 60208. FAX: (708) 491-3728; E-mail: broadbel%wu.edu. 3 FAX: (302) 831-1810; e-mail:
[email protected].
mechanisms generated were entirely equivalent for ethane, propane, and butane when the same carbon count for truncation was used. This size-dependent stopping criterion could thus create some peculiarities in the model. Butane pyrolysis provides an example. A carbon-number stopping criterion of 4 did not allow for subsequent reaction of hexane, a primary product formed from the termination of two propyl radicals formed directly from initiation. However, the formation of acetylene, a quaternary product, was included because of the size of the molecules leading to its formation. The number of insignificant small products increased as the starting alkane increased in length when the only upper bound imposed was the carbon count. The generation of a reaction mechanism for starting alkanes greater than 4 was not possible with a carboncount criterion alone. Because information about all previously generated species must be stored throughout mechanism generation to allow determination of species uniqueness, memory requirements could become prohibitively large. The explosive nature of the reaction mechanism generated for longer chain alkanes consumed all available memory on a NeXT workstation (Motorola 68040 processor, 2.0 MFLOPS, 25 MHz) with 640 MB of available swap space. These issues motivated careful scrutiny of the choice of convergence criteria. Incorporation of a measure of the order in which the products are formed in a network was a logical first step in using chemical and kinetic significance to guide the mechanism termination. Species Rank The order in which a product appears in a reaction mechanism is measured by its product rank (Bhore et al., 1990). The traditional terminology corresponding to the appearance of a product in a reaction network, i.e., primary, secondary, tertiary, is mapped t o a numerical value that is termed its rank. Thus, primary products have a rank of 1, secondary products a rank of 2, and so on. The rank of a product is dependent upon the rank of the reactants that form it. The network shown in Figure 1is used to illustrate these concepts. Product C comes directly from the reactants, A and B,
0888-5885/95/2634-2566$09.00/00 1995 American Chemical Society
Ind. Eng. Chem. Res., Vol. 34,No. 8, 1995 2667
A+B+
C+D+
IF+B+
E
0 Initiation
D
C2H,5
ChainTransfer
0
0
C2H6 + C H 3 -b
0
Figure 1. Reaction network used an example to illustrate concept of species rank.
and thus is primary. Its rank is 1. Another primary product, F, reacts with the reactant B to form D. Formally, the rank of a product formed from reaction of two reactants of different rank has a rank one greater than the maximum of the two reactants' ranks. Thus, the rank of product D formed via the lower reaction pathway is 2. However, product D was also formed as a primary product directly from the bimolecular reaction of A and B. The overall rank of a product formed via several reaction pathways is the minimum of the individual ranks. Because reaction networks generally describe the reaction connectivity of molecular species, the formalism of product ranks was developed accordingly. Chemically, the rank of a species in a network is a measure of the number of molecular intermediates encountered in the species' formation. A radical product will have a rank that is one less than a molecular product formed from the same reaction because it needs yet a t least one final step to be realized as a molecule. Thus, for the present purposes, radical and molecular species were treated according to analogous, yet unique, sets of rules. This chemical logic can be mapped into objective mathematical rules for calculating the rank of molecular and radical network species. The rank of a species in the reaction mechanism is described by the mathematical formalism presented as eqs 1-4. Equations 1and 2
represent the rank of a radical species and a molecular species, respectively, for an individual reaction. The rank of a radical species is equal to the maximum rank of the reactants from which it is formed. The rank of a molecular species is one greater than the maximum reactant rank. The overall product rank of a radical species was determined by the relationship presented as eq 3, and the overall rank of a molecular species was determined by the analogous relationship of eq 4. In both cases, the overall product rank is the minimum of the set comprised of m product ranks, where m is the number of reactions in which the particular species was formed. In eqs 1-4, R is used to denote the rank of radical species; M denotes the rank of molecular species; the subscript p,i denotes that it is a product of reaction i, and the subscript r,i denotes a reactant of reaction i. These ideas are fured through the ethane example shown in Figure 2. The rank of the reactant ethane is 0. The two radicals formed upon initiation of ethane have not passed through any molecular intermediates and are therefore assigned a rank equivalent to the minimum rank of the reactant(s), or 0. The abstraction of hydrogen from ethane by methyl radical forms the first molecular product, methane. Methane is assigned a rank of one greater than the maximum of the two
+
0 M + H*
+
0 H*
H* + C H 3
+
0
0
*CH3 + C H 3
0
1
*C2H, +
n
1
H2
m4 1
+M
+
Figure 2. Reaction mechanism for ethane and the corresponding rank of the species.
reactants'. ranks. In this case, the reactants both have a rank of 0, and therefore, methane is assigned a rank of 1. The assignment of the ranks of the other species shown proceeds in a similar manner.
Product Rank in Convergence Criterion The mechanism generator (Broadbelt et al., 1994) maintains a list of species that are as yet untested for reaction. The reaction mechanism is terminated when this list of untreated species is empty. The placement of a given species in the unreacted species list with the rank-based convergence criterion was dictated solely by its product rank. The species rank, R, or M,, is compared to the overall rank stopping criterion, i, where i is an integer value designating rank. The species does not undergo any subsequent reactions when one of the conditions of eq 5 or 6, dependent on the type of species generated, is met: R, > i
(5)
M,
(6)
>i
In this way, the mechanism generates successive shells of molecular species with increasing product rank. The mechanism presented in Figure 2 is representative of convergence where all primary products were generated but not allowed to undergo subsequent reaction. Further reaction of the primary products results in an array of secondary products and introduces new reaction types. For primary product generation, only bond fission, recombination, H-abstraction, and p-scission occurred. Radical addition did not occur since the only species that possessed the requisite double bond was ethylene, a primary product that was not allowed to react further. Reaction of primary products to only secondary products results in several radical additions to ethylene. A general instance of radical addition is depicted in Figure 3, where a radical derived from a primary product, Rl', adds to the ethylene to form a new radical R2'. Propyl radical formed from initiation of the primary product butane is an example of an R1' radical. Rigorous application of the mathematical formalism of eq 1used to specify product rank assigns a rank 1 to Rz'. If the rank stopping criterion is 1,a newly formed species with
2668 Ind. Eng. Chem. Res., Vol. 34, No. 8, 1995 1
C2H4 I
C2H4 1
C2H4
1
1
+
RI*-
+
R2*
+
1
R2*
Table 1. Output Parameters for Ethane as a Function of Stopping Criteria with the Carbon Count Imposed for All Species
1 4
1
R3*-
R3* 1
R4
... Figure 3. Definition of product rank for radical species results in a n infinite network when addition reactions are included.
rank 1,i.e., Rz', can participate further in reactions with other species. One of the reactions it can undergo is addition t o ethylene. A new radical, R3', is formed that also has a rank of 1. This sequence of subsequent addition reactions without rank increment continues indefinitely. In order to ensure convergence of the reaction mechanism, radical addition, or other reactions that result in no net increase in product rank, was given further consideration. The path forward that was not specific to any reaction type was imposition of an upper bound on the carbon, or heavy atom, count of the products. A species was required to pass both this test and the rank-based criterion to be placed in the unreacted species list for further reaction. The logic of this approach was suggested by the nature of the addition reactions in pyrolysis chemistry. Addition of a C3 radical to ethylene results in formation of a Cg radical. Addition of the Cg radical t o ethylene results in the formation of a C7 radical. Thus, successive addition reactions that do not increment the product rank do increment the number of carbon atoms of the products. A C g radical formed only through addition reactions with ethane as the reactant is formed from three subsequent additions. Imposing an upper bound on the number of carbon caps the number of addition reactions that can take place. The combined convergence criteria resulted in more chemically significant reaction mechanisms for alkane pyrolysis than the carbon-count convergence criterion alone. This can be measured by the number of components, the number of equations, and the time for generation since these are the output parameters that allow comparison of reaction mechanisms. To aid in the discussion of actual network generation results, the specific combination of convergence criteria imposed is designated by ci-rj, where i is the maximum number of carbon atoms a reactive species can contain a n d j is the highest rank of species allowed to react. The symbol r- is used to denote that a carbon-count convergence criterion alone was used. The dependence of the output parameters on the stopping criteria for ethane pyrolysis is presented in Table 1. Each pair of carbon count and rank did not result in a unique reaction mechanism. Several asymptotic limits were apparent. When generation of only primary products was permitted, the reaction mechanism was insensitive t o the carbon count imposed. The resultant network was equivalent to that presented in Figure 2, where no species with a carbon count greater than 2 was generated. Varying the rank criterion at a fured carbon count of 2 revealed no additional reactions or species for ranks greater than 0. The resultant reaction mechanism was equivalent to that for a carbon-
(a) Number of Components 1 2
dr 2 3 4 5 6
0 9 9 9 9 9
dr 2 3 4 5 6
0 37 37 37 37 31
c/r 2 3 4 5 6
0
1
1 1
1 4
1 1 1
12 20 30
=network
11 21 34 47 60
11 47 182 568
3 11 73 606
(b) Number of Equations 1 2 3 55 55 55 190 327 452 412 1770 4058 591 6422 798 (c) Time (s) 2 1
8 59 296
3 1 11 145
4 11
99 94 1
4 55 611 5815
4 1 15 211
(ri) = network (r-)
4 5 6 7 carbon number Figure 4. Comparison of the number of components generated for reaction of ethane with and without a rank-based stopping criterion imposed. Carbon number refers to the carbon-count stopping criterion imposed on all species. 1
2
3
count stopping criterion of 2 alone. There were no species of rank greater than 1that contained two carbon atoms or fewer that could undergo further reaction. The asymptotic behavior of the reaction mechanism is presented more clearly in Figure 4, where the logarithm of the number of components is plotted as a function of the carbon-count stopping criterion, parametric in the rank criterion. The insensitivity of the mechanism to carbon number for generation of primary products only is apparent from the zero slope of the rO line. The large open boxes designate the reaction mechanism generated for ci-r-. Thus, although no upper bound was imposed on the product rank, the carbon-count criteria, c3 and c4, each allowed generation of species up to quaternary rank only. The combination of the two convergence criteria significantly decreased the number of reactions and species that nevertheless captured the chemistry for longer alkane starting reactants. Output parameters for propane pyrolysis modeling as a function of the combined stopping cri6ria are presented in Table 2. The largest number of species that was generated with a carbon-count stopping criterion alone was 941 when c4 was used. Extension of the reaction mechanism t o
Ind. Eng. Chem. Res., Vol. 34, No. 8, 1995 2669 Table 2. Output Parameters for Propane as a Function of Stopping Criteria with the Carbon Count Imposed for All Species dr 3 4 5 6
18 18 18 18
(a) Number of Components 1 2 3 47 73 99 97 341 644 256 2434 617
dr 3 4 5 6
0 129 129 129 129
(b) Number of Equations 1 2 3 327 452 611 1135 2639 4441 3790 18394 10885
dr 3 4 5 6
0 4 3 3 3
0
4 99 941
2 12 89 18033
4 611 5815
%
2
that could be considered. The explosive nature of the reaction mechanism generated for longer chain alkanes with only an upper bound on the carbon count consumed all available memory. Convergence was achieved when the rank-based criterion was included. The output parameters for representative rO reaction mechanisms of n-alkanes up to hexadecane are plotted in Figure 6. A representative set of output parameters is 219 components, 3306 reactions, and 587 s for reaction of hexadecane. The carbon-count stopping criteria was set t o 2n, where n is the number of carbons in the reactant alkane. This ensured that all radicals derived from the reactant would be permitted t o undergo further reaction.
E
1.5 v
-
M
1
3
4
AA
for n-alkanes up to primary products only with a carbon-count stopping criterion equal to 2n, where n is the number of carbon atoms.
4 16 206
3 16 154
i
2
0
carbon number
2
1
2
00
Figure 6. Output parameters for reaction mechanisms generated
y1
2.5
p
with MOPAC
0
*
p & E 8
E
3 2.5
OS0 0-
3
s
3.5 2 v
A
( c ) Time (s)
1 9 36 167 666
---
4
5
6
7
carbon number Figure 5. Comparison of the number of components revealed for different alkane reactants (e = ethane; p = propane; b = butane) with rank and carbon-count stopping criteria imposed.
reaction of species with five carbon atoms was not possible with the memory available. However, with the additional imposition of the rank-based criterion, reactions of species with greater than four carbon atoms were included. For example, reaction of propane with a c5-r2 resulted in a mechanism generated with 2434 species. The additional chemical insight provided by the rankbased stopping criterion was revealed through comparison of reactants of different length with the same carbon-count convergence criterion imposed. The reaction mechanism for ci-rm was insensitive to the length of the starting alkane. This limitation was removed with the introduction of the rank-based constraint. The logarithm of the number of components for different alkane reactants as a function of the carbon-count convergence criterion is illustrated in Figure 5 . A greater number of components, 606, was generated when ethane was reacted to tertiary products with a carbon count of 4, as compared to the reaction of propane to secondary products with the same carbon count, which resulted in 341 species. Certain combinations of reactant, carbon count, and rank result in entirely equivalent reaction mechanisms. For example, the mechanism for butane constrained by a carbon count of 4 and generation of tertiary products was the same as that for ethane with the same carbon count and generation up t o quaternary products. The combination of the two convergence criteria significantly increased the length of the alkane reactant
Carbon Count for Radicals Only The combination of the rank-based and the carboncount-based convergence criteria significantly enhanced the capability of the reaction mechanism generator to apply to larger molecular reactants’ products. However, imposing a carbon-count bound on all types of species resulted in low rank molecular products that were not allowed to react further. The components that were formed from ethane pyrolysis up to secondary products with a carbon count of 3 are presented in terms of their unique identification numbers, overall species rank, and unique string codes in Figure 7. Butane, species 8, is a primary product that was not allowed to react further because of its failure to meet the carbon-count criterion, whereas propane, also a primary product, can undergo subsequent reaction. These species, molecules that meet the rank criterion but do not meet the carboncount criterion, suggested a refinement of the carboncount convergence criterion. The failure of the reaction mechanism to converge with the rank-based criterion alone for pyrolysis chemistry and the culprit addition reactions suggested an appropriate change in the convergence criterion. The buildup in carbon length through successive addition reactions implied that imposing a carbon-countcriterion on only radical species also imposed a higher “rank” on these species without altering the formal mathematical definition of product rank. The lack of a carbon-count bound on molecular species allows them to react further, constrained only by their rank. The notation used to denote the combination of convergence criteria is altered to reflect the application of the carbon count to a specific species type. When the carbon count was imposed on molecules and radicals alike, the stopping criteria are designated by “ci-rj-
2570 Ind. Eng. Chem. Res., Vol. 34, No. 8, 1995 %% --- Component Definitions 1 : 0 : C(H3)C(H3) 2 : 0 : C.(H3) 3 : 0 : C(H3)C.(H2)
4: 1 :C(H4j ' ' 5 : 1 : C(H2)C(H2) 6:O:H.
7 : 1 : C(C(H3)C(H3)H2) 8 : 1 : C(C(H3)H2)C(C(H3)H2) 9 : I : C(C(H3)C.(H2)H2) I O : 1 : C(C(H3)H2)C(C.(H2)H2) 11 : 1 : (H2) 12 : 1 :C.(C(H3)C(H3)H) I3 : 2 : C(C(H2)C(H3)H) 14 : 2 : C(C(C(H3)H2)C(C(H3)H2)H2) 15 : 2 : C(C(C(H3)H2)HZ)C(C(C(H3)H2)H2) 16 : 1 : C(C(C(H3)H2)C(C.(H2)H2)H2) 17 : 2 : C(C(H3)C(H3)C(H3)H) I8 : 2 : C(C(H3)C(H3)H)C(C(H3)H2) 19 : 2 : C(C(C(H3)C(H3)H)C(C(H3)H2)H2) 20 : 2 : C(C(H3)C(H3)H)C(C(H3)C(H3)H) 21 : 1 : C(C(H3)C(H3)H)C(C.(H2)H2) Figure 7. Components formed from ethane pyrolysis allowing up to secondary products to form and species with a carbon count less than or equal to 3 to react. Components are written in the form unique identification number : overall species rank : unique string code representation.
Table 3. Output Parameters for Ethane as a Function of Stopping Criteria with the Carbon Count Imposed for Radical Species Only clr 0 1 2 3 4 5
0 2 4 9 9 9 9
clr 0 1 2 3 4 5
0 1 8
37 37 37 37
(a) Number of Components 1 2 3 2 2 2 4 4 4 13 13 13 22 57 168 34 209 47 (b) Number of Equations 1 2 3 1 1 55 12 12 12 114 114 114 243 958 1711 412 4813 591
4 2 4 13
4 55 12 114
(c) Time (s)
clr 0 1 2 3 4 5
0 0 0 1 1 1 1
1 0 0 3 6 12 20
2
3
4
0 1
0 1
0 1
3 33 241
3 58
3
all". To denote that the carbon count was imposed on radical species alone, "ci-rj-R" is used. Incorporation of this hybrid stopping criterion also resulted in convergent reaction mechanisms. In general, the number of species considered and the number of reactions revealed increased slightly with this less rigid constraint. Output parameters for ethane as a function of stopping criteria with the carbon count imposed for radical species only are presented in Table 3. Asymptotic limits in the reaction mechanism exist for low rank and small carbon counts. For example, 13 species are generated when a carbon count of 2 is imposed for any rank stopping criterion greater than 0.
The additional species and reactions resulting from the altered carbon-count stopping criterion are best revealed through comparison of the two reaction mechanisms. Direct comparison of the number of components as a measure of the reaction mechanism is
0
1
3 4 5 6 7 carbon number Figure 8. Comparison of the number of components for reaction of ethane with the carbon-count stopping criterion imposed for all species (all) and radicals only (R).
2
Table 4. Summary of Eight Models Generated Using Different Convergence Criteria carbon rank carbon count model count criterion applied to 1 0 2 radicals 2 1 2 radicals 2 radicals 3 2 4 3 2 radicals 5 3 0 radicals 6 3 1 radicals 7 3 2 molecules radicals 8 3 molecules radicals
-
notation species reactions cO-r2-R 2 1 cl-r2-R 2 12 c2-r2-R 13 114 c3-r2-R 57 958 c3-rO-R 9 37 c3-rl-R 22 243 c3-r2-all 47 327 c3-rm-all
99
611
provided in Figure 8. In all cases, the number of components was the same or greater for a ci-rj-R mechanism than the corresponding ci-rj-all mechanism. Molecular species that had previously met the rank criterion but whose reactions were truncated because of carbon count were allowed to react with the new application of the carbon-count criterion to radicals only. For example, reaction of ethane for c3-rl-R resulted in the formation of one additional species as compared to c3-rl-all. The molecular species butane is now allowed to participate in subsequent reactions. Abstraction of hydrogen from butane from one of the internal carbons forms a CH~CHZC'HCH~ radical, which is the new species not generated from any previous set of reactions. However, the reaction mechanisms became identical as the carbon count was increased above 3 for a rank of 1. There were no additional primary molecular products of any size larger than four carbons. The difference in the reaction mechanisms increased as the rank criterion was increased. For example, a c3-r2-R mechanism resulted in the formation of 10 more species than found in the c3-r2-all mechanism.
Quantitative Assessment of Convergence Criterion Solution of ethane pyrolysis mechanisms generated using different convergence criteria provided a quantitative measure of the differences in the number of components and equations. Eight different models were generated. The eight models and their associated differences in the stopping criteria are summarized in Table 4. In models 1-4, the carbon-count criterion for radicals only was increased from 0 to 3 at a constant rank cutoff of 2. In models 4-6, the carbon count criterion for radicals only was fixed a t 3, and the rank cutoff was decreased from 2 to 0. Explicitly, the rank
Ind. Eng. Chem. Res., Vol. 34, No. 8, 1995 2671 Table 5. Values of Quantitative Kinetic Parameters Used for Quantitative Comparison of Mechanisms Generated with Different Convergence Criteria reaction family
(A [=I
loglo A Eo L mol-l s-l) (kcal mol-')
s-l,
initiation H-abstraction addition recombination disproportionation p-scission
0 11.5 30.0 0 0 15.0
16.0 8.0 7.5 9.0 7.5 14.0
a 1.0 0.25 0.5 0 0 0.56
Table 6. Generation and Solution Results for Reaction Mechanisms Generated Using Different Convergence Criteria f, fraction number of generation of excess convergence criteria notation ODES time ( 6 ) components 2 1 c0 r2, radicals only cO-r2-R 0 4 0 c l r2, radicals only cl-r2-R 5 13 0 c2 r2, radicals only c2-r2-R 59 57 c3 r2, radicals only c3-r2-R 823 0.193 9 c3 rO, radicals only c3-rO-R 11 0 c3 r l , radicals only c3-rl-R 22 117 0.273 c3 r2, molecules and c3-r2-all 47 281 0.319 radicals c3, molecules and c3-rw-all 99 656 0.374 radicals
cutoff refers to the maximum rank of products that were allowed to react. Thus, only primary products were generated when the rank cutoff was 0, and up to tertiary products were formed for a rank cutoff of 2. These six models were compared to model 7 in which a carbon count of 3 was imposed on molecules and radicals alike with a rank criterion of 2. The final model imposed a carbon count of 3 on both molecules and radicals but imposed no rank cutoff. The latter case is equivalent to the carbon-count convergence criterion applied previously (Broadbelt et al., 1994). Representative values of frequency factors and Polanyi parameters (Nigam and Klein, 1993)for activation energies of Table 5 were used to obtain the rate constant from heat of reaction for each of the members in the elementary step reaction families. The temperature was set to 600 "C, and the initial concentration of ethane was 0.014 mol L-l. The models generated were solved using DASSL (Brenan et al., 19891, a stiff differentialalgebraic equation solver, with a constant convergence tolerance for all cases. Ethane conversion and product yields were predicted up to 150 min of reaction time, where the maximum conversion achieved was 0.373. Results of the generation and solution of the eight different models are reported in Table 6 . The number of ordinary differential equations (ODES)tabulated in the second column is equivalent to the number of species considered in the model. The time for generation is the sum of the time the mechanism generator required to output the reactions and the quantitative values of the reactivity indices and the time for formulation of the balance equations.
Critical evaluation of the quantitative yields of the entire array of species considered in a given model allowed further discrimination among the eight models. The fraction of excess components was defined to quantify the number of species whose reactions did not contribute significantly to reactant disappearance or major product yields. A value of 1.0 x 10-l2 was the cutoff imposed on the yield of molecular products. If a species' yield was never greater than this value throughout the reaction time studied, it was labeled as an excess component. These ideas are captured mathematically in eq 7 by the number fraction of excess components, f , defined as N
N
The output summarized in Table 7 reports the yield of key ethane pyrolysis components after 150 min of reaction. The species reported are ethylene and hydrogen, which are the major products for long-chain kinetics; methane, propane, and butane as representative alkane products; and butadiene, the yield of which is reported to be comparable to that of methane under certain pyrolysis conditions (Plehiers and Froment, 1987). A blank in the component's table entry is used to indicate that the species was not generated given the specific convergence criteria. Inspection of the results in Table 7 reveals that c0r2 and cl-r2 mechanisms fail to capture the kinetics of ethane pyrolysis. The c0-r2 criterion allows only for the formation of methyl radicals. This single carbon species does not meet the carbon-count criterion. Methyl radicals are allowed to recombine to form ethane only as a consequence of generation of all reversible reaction pairs. However, they can participate in no additional reactions. Methyl radicals generated in the cl-rO mechanism abstract hydrogen from ethane t o form methane, but the ethyl radical afforded cannot react further with the carbon count of 1 imposed on radical species. The long-chain pyrolysis products, ethylene and hydrogen, never form in either of these mechanisms. The essentially stoichiometric balance between ethylene and hydrogen and the converted ethane reveal the fast, long-chain kinetics predicted by the remaining mechanisms. Further discrimination among the remaining models requires evaluation of the modeling goals. If accurate prediction of ethylene and hydrogen product yields and ethane conversion are desired, then the streamlined c3-r0 mechanism is sufficient. Conversely, if prediction of butadiene formation is critical, c2-r2, c3-r0, and c3-rl all fail to form butadiene as a product. Thus, these models would be unable to capture the change in the product selectivities observed with a shift in reaction conditions that favor butadiene formation. The three candidate models remaining all impose
Table 7. Yield of Key Components from Ethane Pyrolysis after 150 min of Reaction for Reaction Mechanisms Generated Imposing Different Convergence Criteria cO-r2-R cl-r2-R c2-r2-R c3-r2-R c3-rO-R c3-rl-R c3-r2 -all c3-rm-all
1.0 0.99996 0.641 0.644 0.628 0.641 0.627 0.627
0.359 0.356 0.372 0.359 0.372 0.372
0.359 0.356 0.372 0.359 0.372 0.372
5.79 2.7 9.85 5.79 9.85 9.85
x 10-4
10-4 x 10-4 x 10-4 x 10-4 x 10-4
2.45 x 1.22 x 10-30 8.72 x 10-32
2.61 2.13 2.63 2.11 2.13 2.12 2.12
10-3 x 10-3 x 10-3
10-3 x 10-3 x 10-3
10-4
9.56 x 8.72 10-7 1.04 x 2.96 x 3.23 x 3.22 x lo-@
2672 Ind. Eng. Chem. Res., Vol. 34, No. 8, 1995
a carbon count of 3. These observations suggest that a carbon count of 3, regardless of the types of species to which it is applied, is necessary to describe the pyrolysis of ethane. Additional assessment among the three remaining models was provided by comparing the number fraction of excess components. This was a t its highest value of 0.374 for the model where no rank criterion was imposed. Products that were quaternary or higher were considered in this mechanism but were not formed in significant yield. The lowest fraction of excess components, 0.193, was observed for the c3-r2-R model. Further discrimination was provided by comparing the yields of minor products. Butane was used as an example product. The four carbons of butane are greater than the carbon-count criterion for the c3-r-all or the c3-r2-all mechanisms, both of which impose the carbon count on all species' types. Accordingly, butane does not participate in any significant disappearance reactions. When butane is allowed to react further as in the c3-r2-R mechanism, its yield at 150 min decreased by 75% in comparison to the other two c3 models. In general, allowing reaction of minor primary products has a significant impact on their predicted yield. Overall, a judicious choice of convergence criteria is directed by the reaction conditions and the reaction chemistry. Examination of the results presented for ethane pyrolysis reveal that a rank-based criterion coupled with a carbon-count criterion imposed on only radical species provides the best capability for predicting reactant conversion and major and minor product yields with the fewest number of excess components.
13
q
1
1
dbm
dbm
Inherently Convergent Reaction Mechanisms Application of any convergence criterion is unnecessary for inherently convergent reaction mechanisms. The chemistry is such that exhaustive application of the reaction rules results in a finite number of species. This occurs in all cases when there is no mechanism for molecule growth. Isomerization of substituted alkenes serves as an example of an inherently convergent reaction network. 2-Methyl-1-hexene is a specific member of this class of compounds. The products formed from 2-methyl-1hexene suggest that double bond migration, methyl shifts, and hydrogenation are the dominant reaction pathways. The methyl shifts considered were 1,2methyl shifts across double bonds only. Application of these three reaction operators to the reactant and its progeny results in the reaction network of Figure 9. All species contain 7 carbon atoms and either 14 or 16 hydrogen atoms. There is a finite set of molecules that possess these two empirical formulas. The smaller subset of these molecules that are related through double bond migration, 1,2-methyl shifts, and hydrogenation is also finite. No molecular weight growth occurs, and thus, the reaction network is convergent.
Conclusions The capability t o generate reaction mechanisms via the computer was broadened by introducing a product rank-based convergence criterion. This allowed mechanisms to be generated for reactants and conditions that previously exhausted available memory when only an upper bound of carbon count was imposed. Reaction
Figure 9. Inherently convergent reaction network for 2-methyl1-hexene allowing for double bond migration, 1,a-methyl shifts, and hydrogenation.
mechanisms were successfully generated for n-alkanes up to hexadecane whereas, previously, only reactants of four carbons or fewer were considered. A rank-based convergence criterion alone resulted in the generation of infinite reaction mechanisms when radical addition reactions were considered. The rank-based criterion was combined with an upper bound on the atom count of the radical species to impose a cap on the number of addition reactions that occurred. The coupling of these
Ind. Eng. Chem. Res., Vol. 34, No. 8, 1995 2673 two criteria provided the best capability for predicting reactant conversion and major and minor product yields with the fewest number of excess components. The fraction of excess components was highest of the models studied for the one where no rank criterion was imposed at a value of 0.374. The lowest fraction of excess components, 0.193, was observed for the c3-r2-R model. Isomerization of substituted alkenes served as an example of an inherently convergent reaction network. Inherently convergent reaction networks required no external convergence criteria because there was no mechanism for molecule growth.
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Received for review July 11, 1994 Revised manuscript received November 29, 1994 Accepted December 7, 1994@ I39404339
Abstract published i n Advance ACS Abstracts, J u n e 15, 1995. @
