Two regions

This method, introduced originally in an analysis of nuclear resonance reactions, has been extensively developed [H, 16 and F7] over the past 20 years as a powerful ab initio calculational tool. It partitions configuration space into two regions by a sphere of radius r = a, where r is the scattered electron coordinate. 

In the reaction field method, the space surrounding a dipolar molecule is divided into two regions (i) a cavity, within which electrostatic interactions are sunnned explicitly, and (ii) a surrounding medium, which is assumed to act like a smooth continuum, and is assigned a dielectric constant e. Ideally, this quantity will be... 

Most of the methods proposed include a van der Waals term for describing nonbonded interactions between atoms in the two regions. This is usually represented by a Leonard-Jones 6-12 potential of the form... 

If the QM and MM regions are separate molecules, having nonbonded interactions only might be sufficient. If the two regions are parts of the same molecule, it is necessary to describe the bond connecting the two sections. In most... 

The metal-ion complexmg properties of crown ethers are clearly evident m their effects on the solubility and reactivity of ionic compounds m nonpolar media Potassium fluoride (KF) is ionic and practically insoluble m benzene alone but dissolves m it when 18 crown 6 is present This happens because of the electron distribution of 18 crown 6 as shown m Figure 16 2a The electrostatic potential surface consists of essentially two regions an electron rich interior associated with the oxygens and a hydrocarbon like exterior associated with the CH2 groups When KF is added to a solution of 18 crown 6 m benzene potassium ion (K ) interacts with the oxygens of the crown ether to form a Lewis acid Lewis base complex As can be seen m the space filling model of this... 

Equation 13.23 can also be used as the basis for a curve-fitting method. As shown in Figure 13.14, a plot of In(Ct) as a function of time consists of two regions. At short times the plot is curved since A and B are reacting simultaneously. At later times, however, the concentration of the faster-reacting component. A, decreases to 0, and equation 13.23 simplifies to... 

A preparation of reduced and carboxymethylated protein particles in water reached sedimentation equilibrium after 40 hr at 12,590 rpm. When the data are plotted as suggested by Eq. (9.103), two distinctly linear portions are observed in the graph. The following pairs of points are taken from these two regions ... 

In fossil fuel-fired boilers there are two regions defined by the mode of heat transfer. Fuel is burned in the furnace or radiant section of the boiler. The walls of this section of the boiler are constmcted of vertical, or near vertical, tubes in which water is boiled. Heat is transferred radiatively from the fire to the waterwaH of the boiler. When the hot gas leaves the radiant section of the boiler, it goes to the convective section. In the convective section, heat is transferred to tubes in the gas path. Superheating and reheating are in the convective section of the boiler. The economizer, which can be considered as a gas-heated feedwater heater, is the last element in the convective zone of the boiler. 

The compressors operating range is between two major regions as seen in Fig. 10-74, which is a performance map of a centrifugal compressor. These two regions are. surge, which is the lower flow limit of... 

Naturally, neither of these approximations is valid near the border between the two regions. Physically sensible are only such parameters, for which b < 1. Note that even for a low vibration frequency Q, the adiabatic limit may hold for large enough coupling parameter C (see the bill of the adiabatic approximation domain in fig. 30). This situation is referred to as strong-fiuctuation limit by [Benderskii et al. 1991a-c], and it actually takes place for heavy particle transfer, as described in the experimental section of this review. In the section 5 we shall describe how both the sudden and adiabatic limits may be viewed from a unique perspective. 

There are two regions in the regenerator the dense phase and the dilute phase. At the velocities common in the regenerator, 2-4 ft/sec, the bulk of catalyst particles are located in the dense bed immediately above the air distributor. The dilute phase is the region above... 

A commonly used form of internal modulation is differential reflectometry, in which the reflectance of the sample under investigation (or a portion of it) is compared to a standard material. This can be accomplished either by holding the sample stationary and scanning the probe beam between two regions or by holding the l t spot fixed and moving the sample. ... 

As can be seen from Fig, 3.7, the pinch decomposes the synthesis problem into two regions a rich end and a lean end. The rich end comprises all streams or parts of streams richer than the pinch composition. Similarly, the lean end includes all the streams or parts of streams leaner than the pinch composition. Above the pinch, exchange between the rich and the lean process streams takes place. External MSAs are not required. Using an external MSA above the pinch will incur a penalty of eliminating an equivalent amount of process lean streams from service. On the other hand, below the pinch, both the process and the external lean streams should be used. Furthermore, Fig. 3.7 indicates that if any mass is transferred across the pinch, the composite lean stream will move upward and, consequently, external MSAs in excess of the minimum requirement will be used. Therefore, to minimize the cost of external MSAs, mass should not be transferred across the pinch. It is worth pointing out that these observations are valid only for the class of MEN problems covered in this chapter. When the assumptions employed in this chapter are relaxed, more general conclusions can be made. For instance, it will be shown later that the pinch analysis can still be undertaken even when there are no process MSAs in the plant. The pinch characteristics will be generalized in Chapters Five and Six. 

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