sensitivity

sensitivity - 5.68J/10.652J Spring 2003 Lecture Notes...

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5.68J/10.652J Spring 2003 Lecture Notes Tuesday April 15, 2003 Kinetic Model Completeness We say a chemical kinetic model is complete for a particular reaction condition when it contains all the species and reactions needed to describe the chemical processes at that reaction condition to some specified level of accuracy. In other words, the kinetic model is complete when all the reactions whose rate constants it sets equal to zero really are negligible for the specified reaction conditions and error tolerance. Note that this definition of complete does NOT assume that the rate constants or any other parameters in the model are correct, just that the reactions excluded from the model are indeed negligible. If you believe you can list all possible reactions of the species in the model, a necessary condition for a model to be complete is that the rates of these reactions to form species not included in the model must be smaller than some error tolerance; this approach was introduced by R.G. Susnow et al. (1997). However, this condition is not sufficient: e.g. one of the minor neglected species could be a catalyst or catalyst poison with a very big influence on the kinetics at very low concentrations. If the kinetics are strictly linear you can show that this necessary condition is sufficient (Matheu et al. 2002, 2003); an important example of strictly linear kinetics are the ordinary master equations used to compute pressure-dependent reaction rates. Of course, you usually do not know all the possible reactions. But for purposes of kinetic model reduction one usually assumes that the initial “full” model is complete; you are satisfied if the reduced model reproduces the full model to within some error tolerances under some reaction conditions. Sensitivity Analyses A. Definitions of Sensitivities Suppose you have a model you think is complete at some reaction condition, and you have some estimates for k and Y o so you can numerically solve d Y /dt = F ( Y , k ) Y (t o ) = Y o to get your best prediction for the trajectory Y (t). (Y is normally made up of many mass fractions y i (t) and a few other time-varying state variables e.g. T(t).) You never know the rate (and other) parameters k and the initial concentrations Y o exactly, so you are always interested in how much the predicted trajectory would vary if these values were a little different than the values you assumed, i.e. how sensitive is your prediction to the values of these parameters? A common way to express this is to write Y (t) as a Taylor expansion in the parameters k and Y o :

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y i (t) = y i (t; k,Y o ) + Σ ( y i (t)/ k n )(k n ’– k n ) o + Σ ( y i (t)/ y m )( y o m ’- y o m ) + … The rate constants k usually depend on T and P, so if T and/or P vary with time the first term gets to be a mess. The most popular approach (used in the latest versions of CHEMKIN, but not consistently in earlier versions) is to imagine that all of the rate constants k n (T,P) are multiplied by scaling factors D
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sensitivity - 5.68J/10.652J Spring 2003 Lecture Notes...

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