Equivalent particle

For equivalent particle size the carbon blacks are the most powerful reinforcing fillers. However, fine particle size silicas can be very useful in non-black compounds whilst other fillers such as aluminium hydroxide, zinc oxide and calcium silicate have some reinforcing effect. 

In the previous section we determined the equivalent particle diameter of a set of particles of different sizes, with the aid of which we can treat the mixture as composed of one size of particles, namely The mean free-falling velocity of the mixture is the same. 

This method has been used for the reduction of l-methyl-2-alkyl-.d -pyrrolinium and l-methyl-2-alkyl-.d -piperideinium salts by Lukes et al. (42,249-251) and for the reduction of more complex bases containing the dehydroquinolizidine skeleton by Leonard et al. (252). The formic add reduction may be satisfactorily explained by addition of a hydride ion, or an equivalent particle formed from the formate anion, to the -carbon atom of the enamine (253), as shown in Scheme 13. 

Clearly, the inclusion of the liquid phase (xi) will tend to reduce the number of fines in the system. Thus, a diminution in particle size must be effected to provide an equivalent particle population in the fluid-particle assembly. The particle size ratio normally used in practical systems tends to be somewhat lower than the one computed from Eq. (8). 

Calibration. Many approaches have been used to calibrate flow cytometric measurements. Including the comparison of flow and nonflow techniques (radiolabels, spectrofluorometry). In recent years, commercial standards have been introduced which are calibrated in fluorescein equivalents/particle (e.g., 3,000 or 500,000). With labeled ligands, calibration requires determining the relative quantum yield of the ligand compared to pure fluorescein and using the standards to analyze the amount bound on cells. Our ligands (fluorescein isothiocyanate derivatives) are typically 50% as fluorescent as fluorescein. 

The equivalent particle diameter appearing in these dimensionless groups is the diameter of a sphere having the same external surface area as the particle in question. Thus for a cylinder of length Lc and radius rc, the equivalent particle diameter is given by... 

By comparison of Eqs. (13-19) and (13-20), it is evident that the permeability is identical to the term in brackets in Eq. (13-20), which shows how the permeability is related to the equivalent particle size and porosity of the medium. Since Eq. (13-20) applies only for laminar flow, it is evident that the permeability has no meaning under turbulent flow conditions. 

The catalyst activity depends not only on the chemical composition but also on the diffusion properties of the catalyst material and on the size and shape of the catalyst pellets because transport limitations through the gas boundary layer around the pellets and through the porous material reduce the overall reaction rate. The influence of gas film restrictions, which depends on the pellet size and gas velocity, is usually low in sulphuric acid converters. The effective diffusivity in the catalyst depends on the porosity, the pore size distribution, and the tortuosity of the pore system. It may be improved in the design of the carrier by e.g. increasing the porosity or the pore size, but usually such improvements will also lead to a reduction of mechanical strength. The effect of transport restrictions is normally expressed as an effectiveness factor q defined as the ratio between observed reaction rate for a catalyst pellet and the intrinsic reaction rate, i.e. the hypothetical reaction rate if bulk or surface conditions (temperature, pressure, concentrations) prevailed throughout the pellet [11], For particles with the same intrinsic reaction rate and the same pore system, the surface effectiveness factor only depends on an equivalent particle diameter given by... 

Equation 4.117 makes complete sense. One of the first things one learns in dealing with phase space integrals is to be careful and not over-count the phase space volume as has already been repeatedly pointed out. In quantum mechanics equivalent particles are indistinguishable. The factor n ni is exactly the number of indistinguishable permutations, while A accounts for multiple minima in the BO surface. It is proper that this factor be included in the symmetry number. Since the BO potential energy surface is independent of isotopic substitution it follows that A is also independent of isotope substitution and cannot affect the isotopic partition function ratio. From Equation 4.116 it follows... 

Figure 2. Distribution of selected minerals in Upper Freeport coal as a function of area-equivalent particle diameter (ym).

In a packed bed, the transition between laminar and turbulent flow occurs in the region of Rep 40 Rep is the Reynolds number based on the equivalent particle... 

A represent a case in which the ice bed rides with 10 feet of its height out of the water. The particle diameters refer to equivalent diameters as defined by the Carmen-Kozeney equation which equates particle diameter to the filter properties of a bed. Because small particles give poor filterability, there will be less piston leakage for beds made up of fine particles than for those of coarse particles. Likewise, the drainage properties of the bed from the top to the screen are affected by particle diameter. If it is assumed that the minimum pressure at the screen were to be the same as the pressure above the bed—in other words, full gravity drainage—then the maximum lineal ice rate is established for each equivalent particle diameter. Calculations based on the filtration behavior of the bed and on calorimetric determinations of porosity indicate the approximate relationship ... 

Vmax = maximum lineal ice rate, feet per hour Dp = equivalent particle diameter, inches... 

HPLC column Lichrosorb RP-2 or equivalent, particle size 5 pm mobile phase acetonitrile/0.1 M acetate buffer (pH 4.7) (50 50). 

The problem of symmetry breaking (SB) is well known and multiply discussed in literature. Briefly, we can formulate it as follows. The Hamiltonian of any system of particles forming the Universe is totally symmetric with respect to rotations and reflections in the isotropic space-time, as well as transmutations of identical and equivalent particles, whereas the real objects of the material world composed by these particles do not possess such symmetry. This is seen already from the examples that we live in a world of particles, not antiparticles, and in condensed matter, we have mostly low-symmetry structures. This circumstance can be expressed by the statement that the world is in a state of broken symmetry. An obvious explanation of the contradiction between the totally symmetric Hamiltonian and the broken symmetry of the real world is that the latter is not a solution of its Schrodinger equation. 

Single-particle optical analyzers are especially useful for continuous measurement of particles of uniform physical properties. However, as discussed earlier, uncertainties develop in the measurement of particle clouds that are heterogeneous in composition because the refractive index may vary from particle to particle. Thus, in making atmospheric aerosol measurements, workers have assumed an average refractive index characteristic of the mixture to estimate a calibration curve or have reported data in terms of the equivalent particle diameter for a standard aerosol, such as suspended polystyrene latex spheres. 

The relationship between the catalyst size classification, the equivalent particle diameter and the percentage saving in catalyst or converter volume is summarized in Table 6.174. 

Catalyst Size (mm) Approximate Equivalent Particle Diameter3 (mml Approximate Catalyst Volume Decrease m... 

Particle Size. The equivalent radius R0 may also be reported out as the equivalent particle diameter 2R0. The size may be reported out as mean and standard deviation as shown in Table II or as a histogram as shown in Figure la,b. The size template in Figure 2 illustrates the principle underlying the RQ term. 

The soil material used for the experiments was taken from the field of a former paint factory, a clayey loam with an equivalent particle diameter of 10 pm. It was weathered for more than 20 years and contained 19 wt% hydrocarbons (37 % long alcanes, 34 % monoaromatic, 16% diaromatic, 12% polyaromatic hydrocarbons). Only the agglomerate fraction (hydrocarbons and sand) smaller than 355 pm was employed. As supercritical solvent demineralized water was used. 

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