Null fraction method

The example reactions considered in this section all have the property that the number of reactions is less than or equal to the number of chemical species. Thus, they are examples of so-called simple chemistry (Fox, 2003) for which it is always possible to rewrite the transport equations in terms of the mixture fraction and a set of reaction-progress variables where each reaction-progress variablereaction-progress variable —> depends on only one reaction. For chemical mechanisms where the number of reactions is larger than the number of species, it is still possible to decompose the concentration vector into three subspaces (i) conserved-constant scalars (whose values are null everywhere), (ii) a mixture-fraction vector, and (iii) a reaction-progress vector. Nevertheless, most commercial CFD codes do not use such decompositions and, instead, solve directly for the mass fractions of the chemical species. We will thus look next at methods for treating detailed chemistry expressed in terms of a set of elementary reaction steps, a thermodynamic database for the species, and chemical rate expressions for each reaction step (Fox, 2003). 

The data modeled are from gas chromatograms obtained for Aroclors 1242, 1248, 1254 and 1260. The unknown samples are from the anaysis of used transformer oil obtained from a waste dump in New Jersey. The concentration of individual isomers in selected Aroclor and transformer oil samples are given in Appendix I. The data are organized in a matrix in which the first four data entries for each sample in row 1 of the data array (Table 2, Apendix I) designate the composition of the sample. For standards, these four variables represent the fractional parts of Aroclor 1242, 1248, 1254, or 1260, respectively, that were combined. Results from the analysis of transformer oil (samples 21-23) are of unknown fractional composition and variables 1 through 4 are null entries. In the examples that follow data from samples analyzed (Table 1, Appendix I) were used in part or in total to illustrate the PLS method. 

To perform unbiased analyses, a collaboration-wide policy of blindness was established, where cut selections are optimized on a fraction of data or on time-scrambled data set. We present upper confidence limits for null results following the treatment described in (Feldman and Cousins, 1998) and incorporate systematic uncertainties into the calculation of confidence intervals according to (Conrad et al., 2003). The contributions to systematic uncertainties is predominantly due to variations of the optical properties of the ice, the absolute sensitivity of the OM, the neutrino cross section and the muon propagation. The combined systematic uncertainty is typically 30%, although the value varies slightly with the analysis method. 

The extension of Gillespie s algorithm to spatially distributed systems is straightforward. A lattice is used to represent binding sites of adsorbates, which correspond to local minima of the potential energy surface. The discrete nature of KMC coupled with possible separation of time scales of various processes could render KMC inefficient. The work of Bortz et al. on the n-fold or continuous time MC CTMC) method can lead to computational speedup of the KMC method, which, however, has been underutilized most probably because of its difficult implementation. This method classifies all atoms in a finite number of classes according to their transition probability. Probabilities are computed a priori and each event is successful, in contrast to the Metropolis method (and other null event algorithms) whose fraction of unsuccessful (null) events increases drastically at low temperatures and for stiff problems. In conjunction with efficient search within a class and dynamic variation of atom coordi-nates, " the CPU time can be practically independent of lattice size. After each event, the time is incremented by a continuous amount. 

Alternatively, this same problem can be resolved by spectral subtraction. If a sample of the additive is available, mixtures of the starting material -and product can be separated by subtraction of the spectra of the mixture and its additives. This is achieved by first subtracting the fraction of aspirin, then phenacetin, and finally caffeine, from the unknown sample spectrum to null the corresponding absorptions. The subtraction factors can be read from the spectrometer computer. This second method is of course much faster. 

The hypothetical active site lattice (HASL) method (Doweyko 1988) identifies the points of the network associated to the atoms of the molecule of interest. Then it gives a fraction of the value of the biochemical property to these points. This fraction is characteristic of the analyzed molecule. By repeating the procedure for all the molecules in a given set, some points of the network acquire the summing of the assigned values that are different from null values. These points describe a structure as a map of the active site of the receptor macromolecule that interacts with the effector molecules. 

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