Energy & Fuels 1999, 13, 877-880
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Directed Kinetic Model Building: Seeding as a Model Reduction Tool Prasanna V. Joshi,† Howard Freund,‡ and Michael T. Klein*,†,§ Department of Chemical Engineering, University of Delaware, Newark, Delaware 19716, and Exxon Research and Engineering, Clinton, New Jersey Received December 2, 1998. Revised Manuscript Received April 9, 1999
The need for detailed molecular information from kinetic models has given rise to the practice of modeling the chemistry at either the molecular or mechanistic level. These models are often used to predict the product spectrum of complex process chemistries involving complex feedstocks and, hence, they are extremely large and often consume prohibitively large CPU times. These models therefore need to be tailored to emphasize mainly the important chemistry only. To this end, we have developed a model reduction technique involving the “seeding” of key intermediates and molecules along the important reaction paths in the complex chemistry. This technique directs the model growth toward the most important and experimentally observed products at the expense of the unimportant part of the reaction network. The technique is illustrated using the acid cracking chemistry for n-heptane as an illustrative example. Comparison of a reduced model and a full model reveals considerable time and size savings without loss of accuracy.
Introduction Mathematical models for refinery and petrochemical process chemistries have served the practitioners with assistance in design, revamp, and process control. Enabled by recent advances in analytical chemistry and computational tools, the current industrial trend toward the use of molecule-based kinetic models is motivated by the unprecedented level of molecular detail in the required output. Readily available computer hardware and software allow these models to be quite large. For example, the number of species (equations) in acid cracking and pyrolysis models is often of the order of 500-1000. Thus the construction of the model can be quite time-consuming and expensive. Among the consequences, the fundamental revision of models over time is undermined. Thus, a model tends to be a fixed snapshot in time rather than a “current representation” of available knowledge and ideas. All of these issues have motivated interest in the use of the computer to construct, not just solve, kinetic models. The Mobil SOL approach1,2 provides one elegant example. Other approaches based on Monte Carlo algorithms3 and graph theory4-8 exploit the ability to represent molecules and their reactions on the com* Corresponding author. † University of Delaware. ‡ Exxon Research and Engineering. § Present address: Rutgers, The State University of New Jersey, College of Engineering, B204, 98 Brett Road, Piscataway, NJ 08854. (1) Quann, R. J.; Jaffe, S. B. Ind. Eng. Chem. Res. 1992, 31 (11), 2483. (2) Quann, R. J.; Jaffe, S. B. Chem. Eng. Sci. 1996, 51 (10), 1615. (3) Neurock, M. A Computational Chemical Reaction Engineering Analysis of Complex Heavy Hydrocarbon Reaction Systems. Ph.D. Thesis, University of Delaware, 1992. (4) Baltanas, M. A.; Froment, G. F. Comput. Chem. Eng. 1985, 9, 71-81. (5) Clymans, P. J.; Froment, G. F. Comput. Chem. Eng. 1984, 8, 137-142.
puter. These techniques reduce the model construction burden considerably. For example, an acid crackingrelated kinetic model containing 599 species and 2300 reactions was built in 600 CPU seconds.8,9 In short, the computer can now serve as a “scratch pad” that allows the modeler to realize the implications of chemical and process ideas in terms of code very rapidly. The remaining challenge is to guide this rapid model growth to a finite size. The early phases of model construction will often involve the testing of various chemical concepts. The automated construction of the associated models can expose conceptual problems very rapidly. Often a model will grow to a prohibitively large size, because of the large number of isomers and species that, in principle, can be built. These “divergent” chemistries require that the computer model building algorithm have “external” rules for model termination; otherwise the model will grow large beyond the user constraints, the capability of the CPU memory, or the compiler capacity. In these instances, a strategy for directing the model growth would be desirable. That is, the model growth must discriminate between “feasible reactions” and the finite subset that is kinetically important. The paper describes a strategy for “directed model growth” that exploits user-available empirical knowledge and the concept of product seeding. The essential idea is to direct the model growth to be in the region of important chemistry with regard to the observable (6) Hillewaert, L. P.; Dierickx, J. L.; Froment, G. F. AIChE J. 1988, 34 (1), 17-24. (7) Broadbelt, L. J.; Stark, S. M.; Klein, M. T. Ind. Eng. Chem. Res. 1994, 33, 790-799. (8) Joshi, P. V.; Iyer, S. D.; Klein, M. T. Rev. Proc. Chem. Eng. 1998, 1 (2), 111-140. (9) Joshi, P. V. Molecular and Mechanistic Modeling of Complex Process Chemistries. Ph.D. Dissertation, University of Delaware, 1998.
10.1021/ef980259r CCC: $18.00 © 1999 American Chemical Society Published on Web 05/15/1999
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Figure 1. Dependence of the number of components (N) on the rank (R∞) in the reaction network for different numbers of products per rank (n ) 2, 3, 4).
Figure 2. Dependence of the number of components (N) on the rank (R) in the reaction network for different numbers of seed molecules (s ) 2, 4, 6).
species. We first motivate the problem by considering a limiting-case ideal kinetics model and its associated growth statistics. This motivates the strategy of comparing model growth by (1) starting with only the process chemistry reactant(s), and (2) starting with the process reactant(s) and key products. That is, the process feed stream, or the mathematically nonzero initial conditions of the kinetics model, need not be the model building seed or the starting point for deducing the reaction network. This approach is then using actual examples involving acid cracking chemistry.
increase dramatically, even beyond the point where hardware or software can handle the model. For the present purposes, this size limit is depicted as NC in Figure 1. Beyond NC, the produced model will suffer from RAM and compiler limitations. The example of ethane pyrolysis provides a more chemically relevant illustration of these model growth issues. The yield and composition of aromatic species are important predictions required of a modern cracking model. To fix ideas, assume that, when starting the model building with ethane as the model builder seed, that toluene is a product of rank 10. Figure 1 shows that for n ) 4, the model building will not produce toluene until after NC is reached. Many other mechanistic models containing addition reactions (e.g., radical recombination, radical or carbenium ion addition to an olefin, Diels-Alder reactions) and/or “large n” will behave similarly. This dilemma motivated the model building strategy of product seeding. Graphically, the conceptual idea is to keep the model building process in the “small R, low slope” part of the curves of Figure 1. In the terms of the toluene example, the user seeds the model builder with not only ethane, but also toluene precursors, such as butadiene, methylcyclohexene, etc. More abstractly, the idea is to seed the model builder with the reactant (R ) 0) and R∞/∆R intermediate species. The model building can be allowed to grow to rank increments of ∆R (say ∆R ) 3, for example) for each species. In this way, the model building will be directed along the seed species pathway, and be subdivided into R∞/∆R smaller problems, leading to eq 2.
Background The reaction network is the part of the kinetics model that provides the connectivity among reactant and product molecules. The network can be open or closed in the Boudart10 sense, but will, in general, contain parallel and serial features. An ideal network without loops (or cycles) can be used to illustrate the potential growth in network species with the rank (or generation number, i.e., primary, secondary, etc.) of allowed product species. Consider a chemistry where n (the growth factor) products are formed from the reaction of the initial reactant and all subsequent products. At the start of model building one species will exist. After it reacts to the “primary” products, 1 + n species exist. Primary products (rank ) 1) will react to a secondary (rank ) 2) products, bringing the total to 1 + n + n2. In general, the number of species N as a function of the userspecified final model building rank R∞ will follow the recursive formula shown as eq 1
N)
∑
R
n
∑(n)∆R
N ) (1 + s)
R)R∞
(1)
R)0
The behavior of eq 1 is illustrated in Figure 1 as a plot of N vs R∞ parametric in the growth factor n. Note that N is also the number of species balance equations and thus is a rough measure of the size of the model. Clearly, as n and R∞ increase, the model size can (10) Boudart, M. Kinetics of Heterogeneous Catalytic Reactions; Princeton University Press: Princeton, NJ, 1984.
(2)
where s is the number of seeded intermediate species. Essentially, the computer builds 1 + s smaller (low R) models that, when combined, create a much smaller model than the single, “full” model produced by starting with the process feed only. The behavior of eq 2 is illustrated in Figure 2 for various values of s and R∞/ ∆R for n ) 4. Clearly, this strategy maintains the “low R” growth mode and thus reduces the size of the model for a given R∞ and n.
Seeding as a Model Reduction Tool
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Figure 3. Reaction network for n-heptane in terms of observable species as primary and secondary products. (All the straightchain paraffins and branched olefins are observed as primary products, whereas branched paraffins are observed as secondary products.)
Illustration: Acid Cracking of n-Heptane. Acid Cracking Chemistry. Acid cracking chemistry includes the essential features that motivated the product seeding strategy. Figure 3 summarizes the global chemistry of n-heptane cracking,11 for example. The primary observables are paraffins and olefins; aromatics are higherrank products. The mechanistic reaction network containing the ionic intermediates is considerably more complicated, as there is a large increase in the number of species and reactions with rank. Figure 4 shows the increase in the number of species as a function of the model building termination rank for the case where heptane was the model building feed. In this mechanistic model, both molecular and ionic species are included in the rank count. It is clear that the number of components explodes by R∞ ) 6. The difficulty is that this is below the R∞ needed to build reactions for the formation of aromatics, such as benzene, toluene, and xylenes, from n-heptane. The rank of these aromatic compounds is in the range of seven to nine. Thus, the number of components that would have to be modeled for R∞ ) 8 using n-heptane as the model building feed, from Figure 4, is of the order of 5000. Seeding Strategy. The aim of the seeding strategy is to prune the whole reaction network to its essential building blocks without losing pathways to important products. This will eliminate the unimportant reactions (11) Watson, B. A.; Klein, M. T.; Harding, R. H. Ind. Eng. Chem. Res. 1996, 35, 1506-1516.
Figure 4. Increase in number of species (N) as a function of the termination rank (R) for n-heptane mechanistic reaction network by acid chemistry.
and species. Thus the first step of the seeding strategy is to identify the longest reaction pathway that leads to an important molecule. The second step includes dividing this longest path into various small paths, each of which will “grow” to the termination rank R. The last step amounts to “seeding” the model growth process with the starting species of these small paths as reactants during model construction. For the n-heptane acid cracking illustrative example, the path from n-heptane to toluene was the longest path. In the mechanistic model,11 toluene was a ninthrank product. The path to toluene was therefore divided into four small parts, and the starting species for each
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Table 1. Listing of Seed Molecules in the Reaction Network of n-Heptane Cracking toward the Aromatics (benzene and toluene) Generation Path
Figure 6. Comparison of the reaction network with respect to number of species by seeding the intermediates for nheptane acid cracking. (A reduction of 90% is obtained in the number of species.)
produced identical product spectra and yields, as shown in Figure 5. Figure 6 shows that the seeded model has 90% fewer species. The full model required 30 min of solution time, whereas the seeded model required about 3 min.
Figure 5. Comparison of the yields for models with and without seeding for n-heptane acid cracking. (The model with seeding was terminated after rank three).
of these small paths was included in the initial model building reactant list. Table 1 summarizes this model building “seed vector” of intermediates. The net effect was to direct the growth of the reaction network toward the empirically observed chemistry, including aromatics formation. This in situ model pruning eliminates other higher-rank species not encountered in kinetically significant reaction paths. Comparison of Models’ Predictions. The validity of the seeding strategy is judged in its ability to generate a model that produces the same numerical results as that of a full model. To this end, the computer simulations of the reaction network of n-heptane cracking, with and without11 seeding were compared. Figures 5 and 6 summarize the results obtained using an incremental growth rank of ∆R ) 3. The full and seeded models
Summary and Conclusions A directed model building strategy was devised and successfully used for “in growth” pruning of large, complex chemistry models. Model growth was directed toward empirically observed important products by seeding the model builder with key intermediates. This enabled use of lower growth ranks and kept the final model to within hardware limitations. Two key philosophical points can be made. First, if the goal is to model observable chemistry rather than to predict, from first principles, the identities of the observable products, then the empirical experimental information describing the important reaction products can be a valuable assist in the procurement of a rigorous model. Second, this approach exposes the network construction aspect of model building. Network construction involves the establishment of connections among species, be they reactants or products. The latter designation, while crucial in the practice of a process chemistry, is rather artificial when viewed in the abstract context of a network. Thus the “chronology” of building the network need not be linked in any way to the time evolution of reactants and products in the practice of the process chemistry. The reaction network of n-heptane cracking was used to demonstrate the concept. Aromatic products with a rank greater than seven were accounted for using a seeding strategy with a growth rank of three. The quantitative results of the full and the seeded models were identical, but the seeded model resulted in 90% fewer species and CPU time savings. EF980259R
