Unmixing error

For the processing of spectral data, it is advisable to take only the reference spectra of the fluorophores contained in the sample for the unmixing calculation. Inclusion of unnecessary reference spectra may degrade the resulting image because noise and other aberrations contained in the spectral data would result in a higher unmixing error under these conditions. 

Thus far we have considered only two flow patterns the completely mixed reactor and the completely unmixed reactor. This is because only for these flow patterns can we completely ignore the fluid flow configurations in the reactor. In this chapter we will begin to see how reactors that have more complex flow patterns should be treated. We will not attempt to describe the fluid mechanics completely. Rather, we will hint at how one would go about solving more realistic chemical reactor problems and examine the errors we have been making by using the completely mixed and unmixed approximations. 

More extensive effectiveness charts and relations are available in the literature. The dashed lines in Fig. 11 -26/are for the case of unmixed aud mixed and the solid lines are for the opposite case. The analytic relations for the effectiveness give more accurate re.sults than the charts, since reading errors in charts are unavoidable, and the relations are very suitable for compnt-eri2ed analysis of heat exchangers. 

A second issue is that there is necessarily a trade-off between precision and resolution. That is, if the box sizes are reduced, one obtains more boxes and hence more values of Ca, and consequently any mixing measure will in principle have a lower standard error. On the other hand, this reduction has an obvious limit as the box size approaches the grain size, at which point any blend will statistically appear to be unmixed (because each box can only take on one of the two values, Ca = 0 or Ca = 1). In practice, a happy medium delivering both suitably low standard error and an adequate sample size is easily achieved nevertheless it is important to understand what it is that one is seeking to obtain before deciding on a sampling protocol. 

The reader may detect an apparent error in nomenclature here. Pole 5 for example, is assumed to be a 100 pole and spot 5 on the diffraction pattern is assumed, tacitly, to be due to a 100 reflection. But aluminum is face-centered cubic and we know that there is no 100 reflection from such a lattice, since hkl must be unmixed for diffraction to occur. Actually, spot 5, if our assumption is correct, is due to overlapping reflections from the 200, 400, 600, etc., planes. But these planes are all parallel and are represented on the stereographic projection by one pole, which is conventionally referred to as 100). The corresponding diffraction spot is also called, conventionally but loosely, the 100 spot. 

As shown in Fig. 13.4, the difference in the overall conversion is the consequence of a lower conversion rate at the reactor inlet, as expected. Although the characteristic mixing time corresponds to a distance of 0.24 m, the reaction rate (slope of the conversion profile) of the unmixed feed case approaches that of the premixed feed case by 0.1 m, or 10% of the total reactor volume. Furthermore, the predicted conversion for the immixed and premixed feeds is 25.1% and 25.7%, respectively. This corresponds to a relative difference of 2%, an acceptable level of error that will only decrease as the reactor residence time increases. This analysis validates the reactor design criterion that the characteristic mixing time based on... 

In practice, the selection of the solvent of the right strength can be made either by tedious trial and error method or by using the rapid open bed separation (thin layer chromatography). For strongly adsorbed compounds, it is usually best to use a polar solvent such as an alcohol, pyridine, or an ester, whereas with weakly adsorbed solutes the solvent is normally petroleum ether, carbon tetrachloride or cyclohexane. Mixtures of two or three solvents of different polarity often give better separation than unmixed solvents. 

This factor can be calculated for each specific combination of the natural (analyte) isotope and the enriched (spike) isotope abundances for the element of interest. The description of error propagation plots (see below) includes a number of examples showing how this factor varies as a function of the isotope amount ratios in the spiked sample. The isotope proportions in the unmixed natural and isotopically enriched materials heavily influence the shape of these plots. Thus for... 

It has to be considered that linear unmixing is performed on measured microscope data. Accordingly the values used for the unmixing calculations im-avoidably include measurement errors that depend on the measurement conditions. By using a fitting approach (least squares, see above) vahd results can be obtained even with such deviations and an estimate of the quahty of the fit is possible (for overdetermined but not for determined equation systems). Factors influencing the quahty of linear unmixing will be discussed in the section relevant parameters . 

As shown above, readout noise increases with the number of detection channels used. In cases of significant readout noise, this affects the quality of the unmixing. On the other hand, an estimate of the error of the fit is possible for overdetermined, but not for determined systems by comparing the fit with the raw data of each channel. 

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