One-Dimensional Methods

FRANKEL E N and MEYER A s (2000) The problems of using one-dimensional methods to evaluate multifunctional food and biological antioxidants, JSci FoodAgric, 80,1925-41. 

Here we suggest the steps needed for developing a 2D method. These recommendations can result in either an application that far exceeds a one-column method or an application that fails and is replaced by a one-column method. In the case of a separation that can be adequately resolved with a one-dimensional method, the added... 

The lack of software for the decomposition method in two dimensions in application to electron diffraction texture patterns was avoided by Zvyagin and Zhukhlistov by using one-dimensional methods of intensity extraction. Later the decomposition method for one-dimensional radial profiles (starting at the centre of the diffraction pattern and passing through... 

For larger molecules the situation is different because of the more complicated spectra and because of the smaller NOE enhancements. Two-dimensional inverse-detection experiments are usually the only practical solution to performing relaxation measurements in larger systems, such as biomolecules. In principle, it should be possible to obtain the same signal-to-noise ratio by two-dimensional and one-dimensional methods in... 

Fig. 15.14 Analytical techniques for time-resolved headspace analysis. An electronic nose can be used as a low-cost process-monitoring device, where chemical information is not mandatory. Electron impact ionisation mass spectrometry (EI-MS) adds sensitivity, speed and some chemical information. Yet, owing to the hard ionisation mode, most chemical information is lost. Proton-transfer-reaction MS (PTR-MS) is a sensitive one-dimensional method, which provides characteristic headspace profiles (detailed fingerprints) and chemical information. Finally, resonance-enhanced multiphoton ionisation (REMPI) TOFMS combines selective ionisation and mass separation and hence represents a two-dimensional method. (Adapted from [190])...

Slow chemical exchange in an eight-coordinated bicentered ruthenium complex studied by one-dimensional methods. Data fitting and error analysis. J. Magn. Reson. Ser. A, 118 (1), 21-27. 

In this section we examine the primary transient phenomena that are of interest to SOFC analysis, and provide the fundamental model equations for each one. Examples for the use of these models are given in later sections. While the focus is on reduced-order models (lumped and one-dimensional), depending on the needs of the fuel cell designer, this may, or may not be justifiable. Each fuel cell model developer needs to ensure that the solution approach taken will provide the information needed for the problem at hand. For the goal of calculating overall cell performance, however, it is often that one-dimensional methods such as outlined below will be viable. 

The two-dimensional techniques, both as a preparative tool for collecting fractions and perhaps especially as a separative tool in rapid star electrophoresis, will undoubtedly prove to be of great value in clinical laboratory work. The recent technical improvements in the apparatus allow for easy working conditions. These were previously lacking and delayed the clinical use of methods which are based on a different principle from that of zone electrophoresis. They yield important results in fields where the one-dimensional method has partly failed. 

Modern NMR experiments are used extensively for structure elucidation. A summary of the most important one- and two-dimensional methods for small molecules is given below. The experiments, which can be fully automated and used routinely in academic and industrial laboratories, are subdivided into homo- and heteronuclear one-dimensional methods and homo- and heteronuclear two-dimensional methods (see Friebolin, as above Sanders, J. K. M., and Hunter, B. K., Modern NMR Spectroscopy A Guide for Chemists (Oxford Oxford University Press, 1993)). 

In this respect, there are one-dimensional methods involving collocation that are better classified as two-dimensional methods. 

Since the shape of one of the electrodes is varied during the machining, as is not known in advance, these problems belong to the class of problems with moving (free) boundaries, and their solution involves great difficulties [4-9], Therefore, approximate quasi-steady-state and local, one-dimensional methods, which enable one to reduce the ECM problems to those of known boundaries, are widely used [1-9]. 

Example 13-5 Using the one-dimensional method, compute curves for temperature and conversion vs catalyst-bed depth for comparison with the experimental data shown in Figs. 13-10 and 13-14 for the oxidation of sulfur dioxide. The reactor consisted of a cylindrical tube, 2.06 in. ID. The superficial gas mass velocity was 350 lb/(hr)(ft ), and its inlet composition was 6.5 mole % SO2 and 93.5 mole % dry air. The catalyst was prepared from -in. cylindrical pellets of alumina and contained a surface coating of platinum (0.2 wt % of the pellet). The measured global rates in this case were not fitted to a kinetic equation, but are shown as a function of temperature and conversion in Table 13-4 and Fig. 13-13. Since a fixed inlet gas composition was used, independent variations of the partial pressures of oxygen, sulfur dioxide, and sulfur trioxide were not possible. Instead these pressures are all related to one variable, the extent of conversion. Hence the rate data shown in Table 13-4 as a function of conversion are sufficient for the calculations. The total pressure was essentially constant at 790 mm Hg. The heat of reaction was nearly constant over a considerable temperature range and was equal to — 22,700 cal/g mole of sulfur dioxide reacted. The gas mixture was predominantly air, so that its specific heat may be taken equal to that of air. The bulk density of the catalyst as packed in the reactor was 64 Ib/ft. ... 

From Fig. 13-9 it is apparent that the entering temperature across the diameter is not constant, but varies from a maximum value of about 400°C at the center down to 197°C at the wall. Since the one-dimensional method of solution to be used in this example is based on a uniform temperature radially, we use a mean value of 364°C. 

Design a reactor system to produce styrene by the vapor-phase catalytic dehydrogenation of ethyl benzene. The reaction is endothermic, so that elevated temperatures are necessary to obtain reasonable conversions. The plant capacity is to be 20 tons of crude styrene (styrene, benzene, and toluene) per day. Determine the bulk volume of catalyst and number of tubes in the reactor by the one-dimensional method. Assume that two reactors will be needed for continuous production of 20 tons/day, with one reactor in operation while the catalyst is being regenerated in the other. Also determine the composition of the crude styrene product. 

Even at moderate spinning rates, the anisotropies may be extracted when the slow spinning condition is satisfied. For example, the anisotropy of the interaction can be determined by the simulation of the intensities of the sidebands. This method is simple in principle and in practice because it is a one-dimensional method however, it works optimally only when the spinning speed is at least several times slower than the minimum anisotropy of... 

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