Mathematical modelling sensitivity

Typical approaches for measuring diffusivities in immobilised cell systems include bead methods, diffusion chambers and holographic laser interferometry. These methods can be applied to various support materials, but they are time consuming, making it onerous to measure effective dififusivity (Deff) over a wide range of cell fractions. Owing to the mathematical models involved, the deconvolution of diffusivities can be very sensitive to errors in concentration measurements. There are mathematical correlations developed to predict DeS as... 

A survey of the mathematical models for typical chemical reactors and reactions shows that several hydrodynamic and transfer coefficients (model parameters) must be known to simulate reactor behaviour. These model parameters are listed in Table 5.4-6 (see also Table 5.4-1 in Section 5.4.1). Regions of interfacial surface area for various gas-liquid reactors are shown in Fig. 5.4-15. Many correlations for transfer coefficients have been published in the literature (see the list of books and review papers at the beginning of this section). The coefficients can be evaluated from those correlations within an average accuracy of about 25%. This is usually sufficient for modelling of chemical reactors. Mathematical models of reactors arc often more sensitive to kinetic parameters. Experimental methods and procedures for parameters estimation are discussed in the subsequent section. 

M. Silberbush and S. A. Barber. Sensitivity of simulated phosphorus uptake to parameters used by a mathematical model. Plant Soil 74 93 (1983). 

The calculation of protein proximity and hence association on the basis of sensitized emission or FSPIM requires correction for direct acceptor excitation and donor bleed through using several mathematical models and instrument correction factors [22, 59-61], which is difficult to control [22] (see also Chapters 7 and 8). A high detected acceptor to donor signal ratio in these techniques may also reflect other phenomena than FRET. For instance, this ratio is dependent on cellular expression levels and subcellular localizations, which are difficult to control. Additionally, for the widely used... 

The main tools used to provide global projections of future climate are general circulation models (GCMs). These are mathematical models based on fundamental physical laws and thus constitute dynamical representations of the climate system. Computational constraints impose a limitation on the resolution that it is possible to realise with such models, and so some unresolved processes are parameterised within the models. This includes many key processes that control climate sensitivity such as clouds, vegetation and oceanic convection [19] of which scientific understanding is still incomplete. 

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. 

The Sharpless epoxidation is sensitive to preexisting chirality in selected substrate positions, so epoxidation in the absence or presence of molecular sieves allows easy kinetic resolution of open-chain, flexible allylic alcohols (Scheme 26) (52, 61). The relative rates, kf/ks, range from 16 to 700. The lower side-chain units of prostaglandins can be prepared in high ee and in reasonable yields (62). A doubly allylic alcohol with a meso structure can be converted to highly enantiomerically pure monoepoxy alcohol by using double asymmetric induction in the kinetic resolution (Scheme 26) (63). A mathematical model has been proposed to estimate the degree of the selectivity enhancement. 

Since NMR signal is sensitive to chemical environment, molecular dynamics, and position in space, obtained signal data contain mixture of these types of information. Separation of these types of information often requires detailed analysis using appropriate mathematical model. Improvement in methods and apparatus of NMR may allow us to use more precise mathematical models which are helpful for the separation. 

Step 5 in Table 8-6 involves the computation of the optimum point. Quite a few techniques exist to obtain the optimal solution for a problem. We describe several classes of methods below. In general, the solution of most optimization problems involves the use of a digital computer to obtain numerical answers. Over the past 15 years, substantial progress has been made in developing efficient and robust computational methods for optimization. Much is known about which methods are most successful. Virtually all numerical optimization methods involve iteration, and the effectiveness of a given technique can depend on a good first guess for the values of the variables at the optimal solution. After the optimum is computed, a sensitivity analysis for the objective function value should be performed to determine the effects of errors or uncertainty in the objective function, mathematical model, or other constraints. 

In this paper, we will first illustrate the mathematical models used to describe the coke-conversion selectivity for FFB, MAT and riser reactors. The models also include matrix and zeolite contributions. Intrinsic activity parameters estimated from a small isothermal riser will then be used to predict the FFB and MAT data. The inverse problem of predicting riser performance from FFB and MAT data is straightforward based on the proposed theory. A parametric study is performed to show the sensitivity to changes in coke selectivity and heat of reaction which are affected by catalyst type. We will highlight the quantitative differences in observed conversion and coke-conversion selectivity of various reactors. 

MESI operation requires processing of the whole sample to be extracted and has to reach steady-state permeation, which usually takes a long time. Thus, a new technical modification of MESI, called pulse introduction (flow injection-type) membrane extraction (PIME), has been developed, in which the sample is introduced to the membrane as a pulse pushed by a stream of eluent (usually water).55 This means that attaining a steady state is no longer crucial. PIME therefore provides not only a faster response and higher sensitivity, but also allows extraction of individual samples via discrete injections in addition to continuous on-line monitoring by sequential injection of a series of samples. Guo et al.56 described a mathematical model for the PIME permeation process, which showed that (a) there was a trade-off between the sensitivity and the time lag (the time taken to complete the permeation process) and (b) a large sample volume and a low flow rate enhance the sensitivity but also increase the time lag. 

Following the Morbidelli and Varma criterion, several other methods have been proposed in recent years in order to characterize the highly sensitive behavior of a batch reactor when it reaches the runaway boundaries. Among the most successful approaches, the evidence of a volume expansion in the phase space of the system has been widely exploited to characterize runaway conditions. For example, Strozzi and Zaldivar [9] defined the sensitivity as a function of the sum of the time-dependent Lyapunov exponents of the system and the runaway boundaries as the conditions that maximize or minimize this Lyapunov sensitivity. This has put the basis for the development of a new class of runaway criteria referred to as divergence-based approaches [5,10,18]. These methods usually identify runaway with the occurrence of a positive divergence of the vector field associated with the mathematical model of the reactor. 

This effect is relatively small until the total magnesium ion concentrations reach about 1000 ppm. o The effect of Mg2+ concentration on limestone dissolution rate can be explained by a surface adsorption model. The adsorption of Mg2+ reduces the limestone dissolution rate because the surface is partially blinded by the adsorbed magnesium ions. The competitive adsorption of calcium and magnesium ions was described by a mathematical model based on the Langmuir adsorption isotherm. The model was used to explain the sensitivity of limestone dissolution rate to magnesium ion concentration under limestone DA operating conditions. 

The mathematical model demonstrates the importance of sample collection (r/T), preconcentration (tjc), concentration (C), and detection (k) in a complete trace portal detection system. Of these three subsystems, the detector is the most understood. Considerable information is available that quantifies the sensitivity, specificity, and limits of detection (LOD) for a particular detection method when used for trace explosives detection [13-15], For trace detection portals, the selection of the detection method is based on performance, initial cost, and maintenance issues. The remaining subsystems (sample collection and preconcentration) are the most variable and least understood for their contribution to trace portal performance. Optimizing the explosive removal and transport in sample collection along with preconcentration will enhance the performance of the entire trace detection system. The sensitivity of the detector will help determine the performance needed from the sample collection and preconcentration. 

---