Hyperbolic models estimated parameters

The amount of uncertainty in parameter estimates obtained for the hyperbolic models is particularly large. It has been pointed out, for example, that parameter estimates obtained for hyperbolic models are usually highly correlated and of low precision (B16). Also, the number of parameters contained in such models can be too great for the range of the experimental data (W3). Quantitative measures of the precision of parameter estimates are thus particularly important for the hyperbolic models. (Cl). 

If temperature and the concentrations of all the chemical species in the study were important in both the numerator and denominator of the hyperbolic model, there would be, for n species,(4n+2) unknown parameters in the model. In the example study, there would be 18 parameters. In the "true" expression, there are 10 parameters, because not all species are important in both numerator and denominator. Even if the "true" mechanism were known, there would be too many parameters to estimate simply by tossing the data into a non-linear estimation program. 

Cropley made general recommendations to develop kinetic models for compUcated rate expressions. His approach includes first formulating a hyperbolic non-linear model in dimensionless form by linear statistical methods. This way, essential terms are identified and others are rejected, to reduce the number of unknown parameters. Only toward the end when model is reduced to the essential parts is non-linear estimation of parameters involved. His ten steps are summarized below. Their basis is a set of rate data measured in a recycle reactor using a sixteen experiment fractional factorial experimental design at two levels in five variables, with additional three repeated centerpoints. To these are added two outlier... 

Juma et al. [49] give details of studies to determine the most suitable equation, and the most appropriate method for calculating the values of the equation parameters, for estimating net nitrogen mineralization in the soil. Data obtained for several different types of soil are examined. Both the hyperbolic equation and a first-order equation gave accurate predictions of the amount of nitrogen mineralized over an incubation period of 14 weeks, but the estimates of potentially mineralizable nitrogen and its half-life depended on the model used. 

The determination of more comprehensive coking mechanisms and rate equations requires simultaneous treatment of all experimental data to enable all the relevant parameters related to coking to be considered. After analysing the experimental data, numerical values of the rate and adsorption equilibrium constants were determined by statistical tests, and models were rejected if a negative constant was estimated at more than one temperature. It was found that the hyperbolic type of decay, as described in Equation (1), gives the best fit from the 9 models tested because it gives the least error from the sum of squares analysis [8],... 

The mathematical model forms a system of coupled hyperbolic partial differential equations (PDEs) and ordinary differential equations (ODEs). The model could be converted to a system of ordinary differential equations by discretizing the spatial derivatives (dx/dz) with backward difference formulae. Third order differential formulae could be used in the spatial discretization. The system of ODEs is solved with the backward difference method suitable for stiff differential equations. The ODE-solver is then connected to the parameter estimation software used in the estimation of the kinetic parameters. More details are given in Chapter 10. The comparison between experimental data and model simulations for N20/Ar step responses over RI1/AI2O3 (Figure 8.8) demonstrates how adequate the mechanistic model is. 

The heuristic approach described in this paper utilizes linear statistical methods to formulate the basic hyperbolic non-linear model in a particularly useful dimensionless form. Essential terms are identified and others rejected at this stage. Reaction stoichiometry is combined with the inherent mathematical characteristics of the dimensionless rate expression t< reduce the number of unknown parameters to the critical few that must be evaluated by non-linear estimation. Typically, only four or five parameters remain at this point, and initial estimates are available for these. The approach is equally applicable to cases where the rate-limiting mechanism is known and where it is not. 

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