Optimal diameters

The optimal diameter sizes of the main pipes for each arc of the network... 

Data indicate the presence of a maximum in the performances that is nearly independent on the metal loading. Transmission electron micrographs of the Pd particles indicate that the larger particles (around 15-16 nm) show an irregular, multifaced structure. A more regular structure is observed in the smaller round particles, but apparently with different preferential orientation depending on the dimensions. Note that in the claims of Headwaters Nanokinetix Inc. patents the existence of an optimal diameter for Pd particles is also indicated, to which the preferential exposure ofthe active/selective surface is related. Data in Figure 8.11 do not prove this statement, but are in line with this indication. 

The pseudo parameters of the HSE theory are derived from an equation of state expanded in powers of 1/kT about a hard-sphere fluid, as is developed by the perturbation theory. Consequently, it is reasonable to expect that procedures for defining optimal diameters for the perturbation theory should work well with the HSE procedure. The first portion of this chapter shows that this is indeed correct. The Verlet-Weis (VW) (5) modification of the Weeks, Chandler, and Anderson (WCA) (6) procedure was used here to determine diameters in a mixture of Len-nard-Jones (LJ) (12-6) fluids. These diameters then were used in the HSE procedure to predict the mixture properties. 

We will now discuss the problem of determining effective or optimal diameters for use with the HSE theory for real fluids when both the form of the intermolecular potential and its parameters are unknown but accurate equations of state which represent the PVT behavior over an extensive range are available for the pure components. 

In this case the right side of Equation 27 becomes only Z(p), which is furnished directly by the equation of state as in Equation 30. This is called the high-temperature limit and at some temperature conditions of interest, especially at low densities, the optimal diameters approach it closely. These diameters are always smaller than the high-density limit of Bienkowski and Chao. 

These limits are very nearly upper and lower bounds for the optimal diameters although they do not closely approach the upper bound at any conceivable density of interest. A few cases at low density showed the optimal diameter very slightly below the high temperature limit. The discrepancy is easily within the experimental uncertainty, however. 

The reduced density of 1.6 is considered to be the upper limit of the validity of Equation 36. At densities higher than this 8 and a2 decrease rapidly and a2 itself eventually becomes positive, interpreted as its domination by positive soft-repulsion effects. Diameters from Equation 36 give poor results in this region. There is no way that these soft effects can be separated from attraction effects and the optimal diameter cannot be calculated. 

Equation 38 then is solved for the diameter. This linear fit method gives excellent results for the optimal diameter at reduced densities of about 2.4 and higher. 

Table VI presents preliminary calculations by Chang (24) for a polar-nonpolar mixture. The highest pressures may be invalid because they were made before the method for evaluating the optimal diameters was developed. These computations use the high-temperature limit of the BWR-S equation for ZHs(pd3) to obtain the diameter. It was hoped that comparison with the BWR-S equation would show a more distinct advantage of the theoretical composition dependence of the HSE method. In fact, the two methods give about the same results. Probably no conclusion about this can be drawn from the comparison because the constants determined by Hopke and Lin (25) for the BWR-S equation were obtained by fitting the equation to this binary. The results are given in the column headed BWRSE. The test of the improved composition...

In summary the results show that it is indeed possible to extend the HSE method successfully to mixtures containing polar molecules. Methods have been developed to obtain effective diameters and shape factors which are optimal for use with the HSE theory. Although the determination of diameters for fluids with unknown potential functions with these methods is not possible at all densities, enough calculations can be made to allow a correlation by fitting the results to the VW equations for the optimal diameter with the perturbation theory. The success of the VW diameters for the HSE theory was confirmed. 

We can see from these results that with potentials of 3-5 kV the optimal diameter can be considered to be either 0.35 or 0.5 mm. Since the most nearly constant spot diameter (with changes in potential) was obtained with a needle diameter of 0.35 mm, however, all subsequent tests were performed with a needle electrode of this size. 

By applying this model to a 13-1 bubble-column PBR, Merchuk and Wu found that the optimal diameter of the column (DJ does not change drastically over a wide range of illuminance, as shown in Figure 16a, and is not far from 0.2 m. 

There is a range of volume velocity of two-phase flow, which corresponds to the cone-channel confused with optimal diameter of the diffuser to confuser (further indicated as d /d, ). The distance is limited from bottom by seating stratified two-phase flow, and is limited from top by energy costs arising from the increased pressure on the ends of the device (Dp w ). In particular, the ratio d /d = 3 corresponds to the interval 44 < w < 80 cmVs, and d /d =1.6 corresponds to the interval 80 < w < 180 cm%, and further increase in the velocity of the dispersed system (w > 180 cm%) determines the need to further reduce the ratio d /d imtil d /d = 1,... 

To avoid a large decrease in wavenumber resolution arising from oblique rays, a circular aperture, called the Jacquinot stop (J-stop), is placed in the focal plane of the collimator as shown in Figure 4.12a. The optimal diameter of the J-stop is determined in order to make the wavenumber shift due to oblique rays smaller than the wavenumber resolution 8v determined by the OPD for the beam parallel to the optical axis. 

Infrared radiation from the light source is focused onto the entrance aperture (J-stop), so that any infrared ray with an incident angle a (see Figure 4.12) larger than that determined by the optimal diameter 4 in Equation (4.21) is blocked from advancing to the interferometer. 

Runner length(mm) Optimized Diameter(mm) Thumbrule Diameter(mm)... 

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