Victor S. Engleman
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Detailed Approach to Kinetic Mechanisms in Complex Systems Victor S. Engleman Science Applications, Incorporated, La Joila, Callfornia 92037 (Received May 27, 1977) Publication costs assisted by Science Applications, Incorporated
Much progress has been made in recent years in experimental measurements, critical evaluations, and estimation techniques for rate constants of elementary reaction steps. In many sysems there is sufficient information to take a detailed approach to kinetic mechanisms in complex systems. This paper discusses how one puts together available knowledge to arrive at the simulation for a complex chemical system. It is recommended that the initial judgment be shifted from reaction selection to species selection and that elimination of reactions be made on a rational basis. In the initial calculations, primary attention should be focused on the relative importance of the reactions which are used and on learning whether the results of the calculations make good chemical sense (e.g., are the relative rates of homologous reactions reasonable?). It is also critical to learn which key reactions in the system are not sufficiently well-known and require further study. It is only after these questions are answered that application of the findings to experimental systems can be done with confidence. Specific applications and examples of this approach are discussed briefly.
There has been much recent work with extensive and sometimes massive kinetic models, especially in the combustion field. Mechanisms ranging from tens of reactions to hundreds of reactions have been used to study chemical systems of varying degrees of complexity. The methods used to arrive at these mechanisms are as follows: (a) The fundamental/intuitive approach in which a path from the reactants to the products is developed by selecting reaction paths which are likely to be important-this works best for simple systems. (b) The experiential approach in which mechanisms that have proven successful in the past are used unchanged or extended to include new reaction paths-use of previously successful mechanisms under similar circumstances can prove useful provided the similarity is sufficient; on the other hand adding reaction paths to a previously successful mechanism must be approached with caution because of the possibility of unforeseen interaction between the combined sets. (c) The “extensive” approach in which all reactions which the author can accumulate from the literature are put into the mechanism and the computer is allowed to make the decisions. The approach suggested in this paper combines the extensive approach with the experiential approach with the fundamental/intuitive approach; it does not let the computer establish the final mechanism but rather uses the computer as a tool to extend the capabilities of the scientist while leaving judgment and decisions in his hands. The method suggested is called, for the purposes of this paper, a “detailed” approach to kinetic mechanisms in complex systems. This approach bears similarities to the “extensive” approach mentioned earlier but has several notable additions which bear discussion. While the application of this approach allows objective decisions to be made on mechanisms, in many cases the fundamental/ intuitive approach can give the correct answer in simple systems and provide insight in complex systems. Moreover, the application of the detailed approach does not ensure success as will be discussed later. A distinction should be drawn between studies whose main purpose is to measure the rate of an elementary reaction and those studies that attempt to apply the rates The Journal of Physical Chemistry, Voi. 8 1, No. 25, 1977
of elementary reactions to explain the measurements in a system with multiple parallel pathways. It is the latter of these two applications toward which this paper is directed although many of the principles apply to both applications. The difference in end use and objective is significant, however, since in one case the result is a measurement which adds to the data base of reaction rates and the other is the use of the data base without adding to it. Both applications should recognize the limitations of the existing data base and the dependence of the findings on that base. This paper is concerned primarily with the description of the method and the examples and applications discussed are provided as illustrations of the method and not as the primary aim of the paper. While most of the examples are taken from methane/air combustion, this is not intended to be a paper on that subject, nor is it implied that the chemical system considered is applicable to all possible experiments. However, the approach itself is applicable to all possible experiments with the hazard that it becomes much more difficult and tedious (but more necessary) as the system becomes more complex. The approach can be applied to condensed phase systems and excited state systems although only gas phase systems with equilibrium energy distributions are used as examples. The method of the detailed approach is straightforward and shifts the initial decision from the selection of reactions to selection of species. The steps in the suggested approach are as follows: (1) Selection of species (2) Construction of reactions (3) Evaluation of probable relative importance (4) Survey of kinetic data (5) Evaluation and selection of rates (6) Application These steps will be discussed in the following sections of the paper. Selection of Species The first step is, of course, critical. By selecting too many species the task becomes unnecessarily complicated. By selecting too few species, important reactions can be missed. The species selection should be based on both theoretical considerations and experimental evidence, where available. Thermochemical considerations, reac-
Kinetic Mechanisms in Complex Systems
tants, products, and likely intermediates should all be considered in deciding species to be included. An alternate method might be to establish an intuitive “mechanism” and to see which species were considered; those species would then be used in the following steps. The species selected would depend strongly on the purpose of the calculation and the generality desired in the results. The difficulty of the problem increases exponentially with the number of species considered.
Construction of Reactions The second step is the simplest. It is simply a mathematical construction of all reactions among the selected species. Being a mathematical construction, not all the reactions constructed will make chemical sense, but these bogus reactions will be eliminated in the next step. As an indication of the increasing complexity of the problem with increasing number of species the following is noted. When a given 25 species were selected for methanelair combustion,l322 reactions (and their reverse reactions) were constructed; when 14 species were added to make a total of 39 species, 1078 reactions were constructed. One of the most useful tools from this mathematical exercise is a cross-reference list where all the reactions for a single species are listed together to provide a checklist for the evaluation of the relative importance of competing reactions. The large list of reactions that result from even a small number of species will confirm the need to prescreen the reactions for importance to weed out the impossible reactions or those which are obviously unimportant. Evaluation of Probable Relative Importance The previous steps mark the difference between the recommended approach and the more commonly applied approaches mentioned earlier. If the selection of species has been made properly, the reaction construction step will have provided a shopping list of reactions from which to narrow down a mechanism of significant reactions. In fact, all reactions are occurring a t any moment but the vast majority of them are too slow to be either measurable or significant. Therefore, it is not constructive, instructive, or efficient to include all reactions in a mechanism. It is desirable to eliminate reactions that are not significant for the system being considered and to recognize that a mechanism will not be universally applicable. Even for a seemingly simple system such as methane air combustion, the significant reactions will depend strongly on initial concentrations, dilution by inerts, initial temperature, initial pressure, flow characteristics, initial inhomogeneities, and other configuration-related aspects. Given a set of initial conditions and system characteristics, an evaluation of probable relative importance can be undertaken. Using the cross-reference list, comparisons can be made of all reactions of a given species with the other species considered. Comparison should be made based on thermochemistry, stereochemistry, probable relative concentrations, spin considerations, and other special situations. One should not ignore earlier experience with the same species under different conditions or even different species under similar conditions. Survey and Evaluation of Kinetic Data Ultimately, it is desirable to have accurate information on a large number of elementary reactions over a wide range of conditions (e.g., temperature and pressure). At present there are major inconsistencies in experimental measurements of the rates of some very important reac-
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tions. Therefore, the evaluator is left with the task of assigning the best rates for specific conditions and assigning uncertainties. In too many cases where wellmeaning researchers compare experimental data against a variety of previously proposed mechanisms, is is not clear whether they are comparing reactions or rates from mechanism to mechanism. Therefore, the evaluation step should not be taken lightly and two points should be given careful attention: (1)identify the uncertainty in the selected reaction rates and evaluate the effect of the uncertainty on the calculated result; and (2) strive for consistency in relative rates so that rates of reactions with homologous species make good chemical sense. In the evaluation, it will probably be found that there will be extensive experimental data available for some reactions, limited data available for others, only estimates available for still others, and no information available in the literature for some. Clearly, the assignment of rates and uncertainties is important. It is suggested that in many cases chemical systems will require information on rates which previously have not been measured or estimated. Reasonable rate estimates along with uncertainties in the estimates can be helpful in evaluating whether the reaction might be significant for the chemical system considered. The more complicated the system, the more likely that key intermediate reactions will not have been measured because of the difficulty in their unequivocal measurement. If an estimated reaction is found to be significant in the application, then it may be necessary to find experimental techniques to measure that reaction. The evaluation of best rates is, of course, subject in part to individual judgment, It is desirable to have as much consistency in rate constants from study to study so that when mechanisms are compared it is clear whether we are comparing reactions or rate constants. However, it must be recognized that accurate data are not available on some of the rates and that it is not desirable to perpetuate an inaccurate rate constant for the sake of consistency. On the other hand, it is not desirable to adjust and fit rate constants (either by direct adjustment or by judicious selection of literature data) in complex systems for the sake of matching data. This last point bears further comment. If it can be established, either in simple or complex systems, that a given rate is the critical unknown and that the mechanism is sensitive primarily to its rate, then that provides an indication2 of the rate of that reaction, or at least of the importance of that reaction and the need for measurement of its rate. On the other hand, if several rates require adjustment or if one or more of the rates require adjustment outside of carefully considered rate uncertainties, then that is an indication that the experiment or the mechanism or several rate constants require further study. The matter of agreement between calculation and experiment will be discussed further in the subsequent section. Calculations of chemical mechanisms are aided by a growing body of critically evaluated rate constants. Critical evaluations are a valuable starting point and provide a basis for consistent calculations. If they also include survey information they provide a valuable guide to the literature. However, it should be recognized that many critical surveys clearly limit the range of conditions over which the rates are considered valid, and extrapolation beyond those conditions should be approached with proper caution. In addition, many critical surveys indicate uncertainties in the rate constants evaluated; these uncertainties should not be ignored. These uncertainties can result from experimental circumstances, from disagreement in meaThe Journal of Physical Chemistry, Vol. 8 l , No. 25, 1977
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surements by different investigators, or from attempting to apply a single rate expression across a wide range of conditions. The critical survey is no better than the experimental data or estimates that it is based on so that new information on the rates of individual reactions should be assessed. In addition, it should always be recognized that the rates represent chemical reactions and not merely mathematical expressions and are therefore subject to judgments that go beyond the calculated result for the complex system. The individual reactions and their rates must make good chemical sense. The applications must give careful consideration to this fact as discussed in the next section. The fundamental points to remember are the following: (1)assemble the best available rates before you start the calculations, (2) do not be so firmly committed to your rates that you neglect to evaluate whether they make good chemical sense for your application, (3) do not be so loosely committed to your rates that you fall into the trap of using the rates as adjustable parameters to fit experimental data.
Applications It is suggested that first pass calculations be used to identify any obvious mistakes in estimated rates. Even though the survey and evaluation may have resulted in reasonable rates and uncertainties, it is possible that the relative rates of comparable reactions may be out of proportion. Generally, the first round calculation will indicate where this is the case if the investigator looks for it. For example, in the case of hydrocarbon combustion reactions with air, the relative rates of reaction of hydrocarbon fragments such as CH3, CH2 with a given reactant and the relative rates of a given intermediate with OH, H, and 0 should be compared for consistency and the reason for differences should be investigated. The second pass calculations should be used to identify which reactions are probably the most important for the application. It is possible that the first pass calculation has provided a good deal of insight into which reactions are important. Obtaining this information requires a computer program that provides either diagnostic information on the relative amounts of a given intermediate that reacts by a given path or some other indication of the sensitivity of the calculation to a given reaction. The key is to learn what the calculations tell you; look for interpretations rather than matching of experimental data. The above suggestions should not restrict the use of experimental data to provide calibration for the model but rather should direct the user away from obtaining best fit calculations and toward obtaining useful chemical information on the complex chemical system. If it is known which reactions are most important and good rates are available for those reactions a good fit will be obtained. However, it must be recognized that currently not enough
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S.Engleman
good rate information is available to solve every chemical system. Therefore, the most useful information that can come out of chemical kinetics calculations on complex systems is the following: (a) Identification of important reactions (b) Identification of reactions needing further study (e) Identification of key experiments The above information will be useful to future investigators who may have a similar system which differs somewhat from a system already studied. If a 30- or 300-step mechanism is established by intuition, and it is shown that it matches experimental data without indicating which reactions were the most significant, that has limited utility. If the 30-step mechanism is a distillation of a 100-step mechanism by retaining only significant reactions, that is more useful. If the reader can be instructed about the important reactions for key paths in the mechanism from reactants to products and where the uncertainties in the rates limit the accuracy of the calculations, that is even more useful. As mentioned earlier, all possible reactions are occurring simultaneously and it is up to the investigator to identify which are the most important for his particular case and to identify them in such a way that it will be useful to future investigators. Thus, the application should be made from the point of view of contributing to the general knowledge and not from the point of view that it will establish a mechanism of universal applicability. A third pass calculation, if necessary, should be used to identify how closely the mechanism matches the experimental results and where (1)additional chemical kinetics information is needed, and (2) additional experiments or measurements are needed to test the calculations of the mechanism. The detailed approach can be very useful if properly executed. It consists of the following steps as discussed in this paper: (1) Selection of species (2) Construction of reactions (3) Evaluation of probable relative importance (4) Survey of kinetic data (5) Evaluation, selection, and estimation of rates (6) Application to problem (a) Identify important reactions (b) Identify rate information needed (c) Identify key experiments (d) Calibrate with experiment without arbitrary adjustment
References and Notes (1) V. S. Engleman, "Survey and Evaluation of Kinetic Data on Reactions in Methane/Air Combustion", EnvironmentalProtection Agency Report No. EPA-600/2-76-003, Jan 1976. (2) In complex systems one is always faced with the possibility that agreement with experiment is fortuitous.
