Kinetic model sensitivity analysis

K. Radhakrishnan, Combustion Kinetics and Sensitivity Analysis Computations, in Numerical Approaches to Combustion Modelling 83-128, Prog, in Astronaut. Aeronaut. 135 (AIAA, Washington, 1990). 

Brandt and co-workers proposed a detailed mechanism for this reaction which was validated using kinetic modeling and the most viable values of the rate constants were estimated on the basis of sensitivity analysis (80). In this model, the absorbance increase observed at the... 

Finally, we should mention that in addition to solving an optimization problem with the aid of a process simulator, you frequently need to find the sensitivity of the variables and functions at the optimal solution to changes in fixed parameters, such as thermodynamic, transport and kinetic coefficients, and changes in variables such as feed rates, and in costs and prices used in the objective function. Fiacco in 1976 showed how to develop the sensitivity relations based on the Kuhn-Tucker conditions (refer to Chapter 8). For optimization using equation-based simulators, the sensitivity coefficients such as (dhi/dxi) and (dxi/dxj) can be obtained directly from the equations in the process model. For optimization based on modular process simulators, refer to Section 15.3. In general, sensitivity analysis relies on linearization of functions, and the sensitivity coefficients may not be valid for large changes in parameters or variables from the optimal solution. 

Since detailed chemical kinetic mechanisms involve the participation of a large number of species in a large number of elementary reactions, sensitivity and reaction path analyses are also essential elements of DCKM. Sensitivity analysis provides a means to assess the limits of confidence we must put on our model predictions in view of uncertainties that exist in reaction rate parameters and thermochemical and thermophysical data utilized, as well as the initial and boundary conditions used in the modeling work. Through... 

In Fig. 1, various elements involved with the development of detailed chemical kinetic mechanisms are illustrated. Generally, the objective of this effort is to predict macroscopic phenomena, e.g., species concentration profiles and heat release in a chemical reactor, from the knowledge of fundamental chemical and physical parameters, together with a mathematical model of the process. Some of the fundamental chemical parameters of interest are the thermochemistry of species, i.e., standard state heats of formation (A//f(To)), and absolute entropies (S(Tq)), and temperature-dependent specific heats (Cp(7)), and the rate parameter constants A, n, and E, for the associated elementary reactions (see Eq. (1)). As noted above, evaluated compilations exist for the determination of these parameters. Fundamental physical parameters of interest may be the Lennard-Jones parameters (e/ic, c), dipole moments (fi), polarizabilities (a), and rotational relaxation numbers (z ,) that are necessary for the calculation of transport parameters such as the viscosity (fx) and the thermal conductivity (k) of the mixture and species diffusion coefficients (Dij). These data, together with their associated uncertainties, are then used in modeling the macroscopic behavior of the chemically reacting system. The model is then subjected to sensitivity analysis to identify its elements that are most important in influencing predictions. 

Laidler, 1987). However, their use in the development of detailed chemical kinetic models is seldom justified because of the complexities involved in the calculations, and the need accurately to know data that are frequently unknown or difficult to estimate. In addition, such calculations may be unnecessary if the associated reactions subsequently are determined to be unimportant by sensitivity analysis. 

Equation (48) e ees with experimental results in some circumstances. This does not mean the mechanism is necessarily correct. Other mechanisms may be compatible with the experimental data and this mechanism may not be compatible with experiment if the physical conditions (temperature and pressure etc.) are changed. Edelson and Allara [15] discuss this point with reference to the application of the steady state approximation to propane pyrolysis. It must be remembered that a laboratory study is often confined to a narrow range of conditions, whereas an industrial reactor often has to accommodate large changes in concentrations, temperature and pressure. Thus, a successful kinetic model must allow for these conditions even if the chemistry it portrays is not strictly correct. One major problem with any kinetic model, whatever its degree of reality, is the evaluation of the rate cofficients (or model parameters). This requires careful numerical analysis of experimental data it is particularly important to identify those parameters to which the model predictions are most sensitive. 

The first two sections of Chapter 5 give a practical introduction to dynamic models and their numerical solution. In addition to some classical methods, an efficient procedure is presented for solving systems of stiff differential equations frequently encountered in chemistry and biology. Sensitivity analysis of dynamic models and their reduction based on quasy-steady-state approximation are discussed. The second central problem of this chapter is estimating parameters in ordinary differential equations. An efficient short-cut method designed specifically for PC s is presented and applied to parameter estimation, numerical deconvolution and input determination. Application examples concern enzyme kinetics and pharmacokinetic compartmental modelling. 

P. Glarborg, R.J. Kee, and J.A. Miller. Kinetic Modeling and Sensitivity Analysis on Nitrogen Oxide Formation in Well Stirred Reactors. Combust. Flame, 65 177-202, 1986. 

There are many tricks and shortcuts to this process. For example, rather than compiling all of the transformation rate equations (or conducting the actual kinetic experiments yourself), there are many sources of typical chemical half-lives based on pseudo-first-order rate expressions. It is usually prudent to begin with these best estimates of half-lives in air, water, soil, and sediment and perform a sensitivity analysis with the model to determine which processes are most important. One can return to the most important processes to assess whether more detailed rate expressions are necessary. An illustration of this mass balance approach is given in Figure 27.5 for benzol a Ipyrene. This approach allows a first-order evaluation of how chemicals enter the environment, what happens to them in the environment, and what the exposure concentrations will be in various environmental media. Thus the chemical mass balance provides information relevant to toxicant exposure to both humans and wildlife. 

There are also two factors that have already been noted in the numerical analysis of the kinetic model of CO oxidation (1) fluctuations in the surface composition of the gas phase and temperature can lead to the fact that the "actual multiplicity of steady states will degenerate into an unique steady state with high parametric sensitivity [170] and (2) due to the limitations on the observation time (which in real experiments always exists) we can observe a "false hysteresis in the case when the steady state is unique. Apparently, "false hysteresis will take place in the region in which the relaxation processes are slow. 

Often there are cases where the submodels are poorly known or misunderstood, such as for chemical rate equations, thermochemical data, or transport coefficients. A typical example is shown in Figure 1 which was provided by David Garvin at the U. S. National Bureau of Standards. The figure shows the rate constant at 300°K for the reaction HO + O3 - HO2 + Oj as a function of the year of the measurement. We note with amusement and chagrin that if we were modelling a kinetics scheme which incorporated this reaction before 1970, the rate would be uncertain by five orders of magnitude As shown most clearly by the pair of rate constant values which have an equal upper bound and lower bound, a sensitivity analysis using such poorly defined rate constants would be useless. Yet this case is not atypical of the uncertainty in rate constants for many major reactions in combustion processes. 

The optimized parameters for the kinetic model used to describe the catalytic reactions of the three oxygenates are listed in Table VII. In this analysis, eight parameters were found to be sensitive, and the 95% confidence limits are given for these parameters in Table VII. The solid curves in Figs. 11-13 represent predictions of the kinetic model under various reaction conditions. Good agreement is achieved between the predictions of the model and the experimental reaction kinetics for all three reactions. 

The kinetic and deactivation models were fitted by non-linear regression analysis against the experimental data using the Modest software, especially designed for the various tasks -simulations, parameter estimation, sensitivity analysis, optimal design of experiments, performance optimization - encountered in mathematical modelling [6], The main interest was to describe the epoxide conversion. The kinetic model could explain the data as can be seen in Fig. 1 and 2, which represent the data sets obtained at 70 °C and 75°C, respectively. The model could also explain the data for hydrogenated alkyltetrahydroanthraquinone. 

When once a mechanism has been built up, techniques of sensitivity analysis (see Sect. 2.5.4) and of parametric estimation (see Sect. 5) allow a determination both of the numerical values of a few kinetic parameters (or of combinations thereof) and the degree of confidence which can be placed in these estimates (assuming, as usual, that there are no systematic errors both in experimental results and in reaction and reactor models). 

In order to interpret a wide range of combustion problems over a wide range of conditions, advances in understanding have of necessity had to be made in the theories of kinetics and dynamics, modelling and sensitivity analysis, the interplay of chemical and physical effects, and the interactions of both of these with thermal factors. The later chapters will focus on these and other topics. 

The family of methods for the study of parametric information in mathematical models is called sensitivity analysis. Sensitivity analysis investigates the relationship between the parameters and the output of any model. It is usually used for two purposes first, for uncertainty analysis and, second, for gaining insight into the model. Sensitivity methods in chemical kinetics have been reviewed by Rabitz et al. [64], who concentrated mainly on the interpretation of sensitivity coefficients in reaction-diffusion systems. Turdnyi [12] considered sensitivity methods as tools for studying reaction kinetics problems and reviewed several applications. Recently, Radha-... 

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