Identifiability problem, Kinetic parameter

Once the theoretical yield from a crystallizer has been calculated from mass and energy balances, there remains the problem of estimating the CSD of the product from the kinetics of nucleation and growth. An idealized crystallizer model, called the mixed suspension-mixed product removal model (MSMPR), has served well as a basis for identifying the kinetic parameters and showing how knowledge of them can be applied to calculate the performance of such a crystallizer, ... 

In addition, when the inverse problem is solved, namely the volume-average rates and kernels are identified by comparison between experiments and model predictions under the assumption of perfect mixing (or, in other words, by using Eq. (7.146)), one must remember that the volume-average rates and kernels are not truly kinetic parameters but contain some fluid-dynamic factors in them. This is why, on transferring these rates and kernels to different operating conditions or from one system to another, very poor performances can be observed. 

The kinetic parameters in this form of the rate law can be identified with the slopes (gii, Si2> and g,3) and intercqit (In a,) in a linear coordinate system relating the logarithm of V, to the logarithms of the X,. Estimating the values for these kinetic parameters from appropriate experimental data is a solvable problem in linear-regression (see above). This is in sharp contrast to most other nonlinear formalisms for which there are no general methods that are practical for extracting kinetic parameters from experimental data (see above). 

To introduce the idea of identifiability, we start with two very simple problems, one from enz3rme kinetics and one from compartmental modeling. In the next section, we classify the parameters section IV then gives a general statement of the identifiability problem. 

Which NH2 source to use depends on the kinetic problem and the specific facilities available. In flow reactors, heterogeneous depletion of NH2 and other intermediate radicals can not be avoided, thus giving erroneous kinetic parameters, if the y values (as defined above) depend on the reactant concentrations. The photolytic NH2 production is disadvantageous, if the reactant also absorbs the photolysis light in the case of 1,3-butadiene [127], for example, the reaction of excited species or of photofragments may interfere. NH2 production in flames is part of a complex system here it is difficult to identify an elementary process, for which the kinetic parameters are to be determined. 

It is important to distinguish clearly between the surface area of a decomposing solid [i.e. aggregate external boundaries of both reactant and product(s)] measured by adsorption methods and the effective area of the active reaction interface which, in most systems, is an internal structure. The area of the contact zone is of fundamental significance in kinetic studies since its determination would allow the Arrhenius pre-exponential term to be expressed in dimensions of area"1 (as in catalysis). This parameter is, however, inaccessible to direct measurement. Estimates from microscopy cannot identify all those regions which participate in reaction or ascertain the effective roughness factor of observed interfaces. Preferential dissolution of either reactant or product in a suitable solvent prior to area measurement may result in sintering [286]. The problems of identify-... 

Although the investigations of both Raunkjaer et al. (1995) and Almeida (1999) showed that removal of COD — measured as a dissolved fraction — took place in aerobic sewers, a total COD removal was more difficult to identify. From a process point of view, it is clear that total COD is a parameter with fundamental limitations, because it does not reflect the transformation of dissolved organic fractions of substrates into particulate biomass. The dissolved organic fractions (i.e., VFAs and part of the carbohydrates and proteins) are, from an analytical point of view and under aerobic conditions, considered to be useful indicators of microbial activity and substrate removal in a sewer. The kinetics of the removal or transformations of these components can, however, not clearly be expressed. Removal of dissolved carbohydrates can be empirically described in terms of 1 -order kinetics, but a conceptual formulation of a theory of the microbial activity in a sewer in this way is not possible. The conclusion is that theoretical limitations and methodological problems are major obstacles for characterization of microbial processes in sewers based on bulk parameters like COD, even when these parameters are determined as specific chemical or physical fractions. 

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. 

In LSV experiments at stationary electrodes, there can be unwanted effects due to natural convection forced convection and a uniformly accessible electrode obviate this problem. The minimum voltage scan rate at which LSV effects appear (i.e. steady-state assumptions fail) will depend on the electrode kinetics and flow parameters. We can immediately identify two extreme situations. 

A reverse kinetic problem consists in identifying the type of kinetic models and their parameters according to experimental (steady-state and unsteady-state) data. So far no universal method to solve reverse problems has been suggested. The solutions are most often obtained by selecting a series of direct problems. Mathematical treatment is preceded by a qualitative analysis of experimental data whose purpose is to reduce drastically the number of hypotheses under consideration [31]. 

Identification. In most cases, the mathematical models of interest in industry contain a few parameters whose values, essentially unknown a priori, must be computed on the basis of the available experimental data. In the case considered here, chemical kinetics is the main field in which this problem is of concern. Identification provides methods for obtaining the best estimates of those parameters and for choosing (i.e., identifying) the best mathematical model among different alternatives. 

Numerical identifiability also becomes a problem with a poorly or inadequately designed experiment. For example, a drug may exhibit multi-exponential kinetics but due to analytical assay limitations or a sampling schedule that stops sampling too early, one or more later phases may not be identifiable. Alternatively, if sampling is started too late, a rapid distribution phase may be missed after bolus administration. In these cases, the model is identifiable but the data are such that all the model components cannot be estimated. Attempting to fit the more complex model to data that do not support such a model may result in optimization problems that either do not truly optimize or result in parameter estimates that are unstable and highly variable. 

The simple carrier of Fig. 6 is the simplest model which can account for the range of experimental data commonly found for transport systems. Yet surprisingly, it is not the model that is conventionally used in transport studies. The most commonly used model is some or other form of Fig. 7. In contrast to the simple carrier, the model of Fig. 7, the conventional carrier, assumes that there exist two forms of the carrier-substrate complex, ES, and ES2, and that these can interconvert by the transitions with rate constants g, and g2- Now, our experience with the simple- and complex-pore models should lead to an awareness of the problems in making such an assumption. The transition between ES, and ES2 is precisely such a transition as cannot be identified by steady-state experiments, if the carrier can complex with only one species of transportable substrate. Lieb and Stein [2] have worked out the full kinetic analysis of the conventional carrier model. The derived unidirectional flux equation is exactly equivalent to that derived for the simple carrier Eqn. 30, although the experimentally determinable parameters involving K and R terms have different meanings in terms of the rate constants (the b, /, g and k terms). The appropriate values for the K and R terms in terms of the rate constants are listed in column 3 of Table 3. Thus the simple carrier and the conventional carrier behave identically in... 

Adsorption kinetics of a single particle (activated carbon type) is dealt with in Chapter 9, where we show a number of adsorption / desorption problems for a single particle. Mathematical models are presented, and their parameters are carefully identified and explained. We first start with simple examples such as adsorption of one component in a single particle under isothermal conditions. This simple example will bring out many important features that an adsorption engineer will need to know, such as the dependence of adsorption kinetics behaviour on many important parameters such as particle size, bulk concentration, temperature, pressure, pore size and adsorption affinity. We then discuss the complexity in the dealing with multicomponent systems whereby governing equations are usually coupled nonlinear differential equations. The only tool to solve these equations is... 

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