Particle-size distribution dimensionless

Figure 18 is an entrainment or gas-carryiag capacity chart (25). The operating conditions and particle properties determine the vertical axis the entrainment is read off the dimensionless horizontal axis. For entrainment purposes, the particle density effect is considered through the ratio of the particle density to the density of water. When the entrainable particle-size distribution is smaller than the particle-size distribution of the bed, the entrainment is reduced by the fraction entrainable, ie, the calculated entrainment rate from Figure 18 is multipfled by the weight fraction entrainable. 

When Equation 9 is used in Equation 8 along with the relationships for the residence time distributions one obtains the following dimensionless particle size distributions for one- and two-tank systems. 

PSD Dimensionless particle size distribution Re Reynolds number... 

Often, the same expressions from this model are applied to the reductions of pellets, in which cases such structural factors as particle size distribution, porosity and pore shape, its size distribution, etc. should really affect the whole kinetics. Thus, the application of this model to such systems has been criticised as an oversimplification and a more realistic model has been proposed [6—12,136] in which the structure of pellets is explicitly considered to consist of pores and grains and the boundary is admitted to be diffusive due to some partly reduced grains, as shown in Fig. 4. Inevitably, the mathematics becomes very complicated and the matching with experimental results is not straightforward. To cope with this difficulty, Sohn and Szekely [11] employed dimensional analysis and introduced a dimensionless number, a, given by... 

When the extinction is measured at different frequencies/, this equation becomes a linear equation system, which can be solved for Cpp and q2(x). The key for the calculation of the particle-size distribution is the knowledge of the related extinction cross section K as a function of the dimensionless size parameter c = iTOifk. For spherical particles K can be evaluated directly from the acoustic scattering theory. A more general approach is an empirical method using measurements on reference instruments as input. 

The linear normal distribution (bell curve Gaussian normal distribution ) is generally suitable for very narrow particle-size distributions. The standardized, dimensionless shape of the normal distribution produces a straight line on semi-logarithmic probability paper. 

This chapter has presented a theoretical derivation of continuous particle size distributions for a coagulating and settling hydrosol. The assumptions required in the analysis are not overly severe and appear to hold true in oceanic waters with low biological productivity and in digested sewage sludge. Further support of this approach is the prediction of increased particle concentration at oceanic thermoclines, as has been observed. This analysis has possible applications to particle dynamics in more complex systems namely, estuaries and water and waste-water treatment processes. Experimental verification of the predicted size distribution is required, and the dimensionless coeflBcients must be evaluated before the theory can be applied quantitatively. 

In order to circumvent the problem of unwieldy numerical fractions having several digits (e.g., the clay-silt bonndary at 0.001953mm) and to simplify (at that time) the laborious hand calculations of various statistical parameters for the particle size distributions, Krumbein (1934) subseqnently proposed that the size of particles should be measured in dimensionless units of 0 (phi). Phi is defined as ... 

Deconvolution of particle size distribution from dimensionless moments. 

If the initial particle size distribution is monodisperse with dimensionless size unity, then g(s) = e and the transform (4.3.11) may be inverted by using the expansion of (1 — y) in powers of y, which converges for y less than unity. Thus... 

Where (p is the particle sphericity and PSD is the dimensionless particle size distribution. 

Glicksman et al. (1995) used the simplified scaling parameters to construct a one-half linear scale model of a Foster Wheeler circulating bed combustor pressurized to 14 bar. The combustor has a 20.3 cm inner diameter with an overall height of 8.3 m with both a primary and a secondary air supply. The solids recycle rate was accurately determined by a calorimetric balance of a fluidized bed heat exchanger in the return loop of the circulating bed. The cold model, one-half scale, used polyethylene plastic particles to match the dimensionless particle size distribution as well as the gas-to-solid density ratio. 

One of the interesting, and perhaps unexpected, results of the CSTR model studies is the nature of predicted particle size distributions. Figure 6 shows the influence of radical desorption from the latex particles (the parameter y) on the effluent latex PSD. The distributions are plotted in terms of a dimensionless diameter d/(6v /tt) /. ... 

FIG. 7.50. Plot of extent of reaction aganst dimensionless reactor residence time for spherical particles made up of spherical grains in a batch or plug flow reactor. Particle size distribution given by the Gates-Gaudin-Schumann equation. Fg = 3. 

Roy et al. (R3) define the critical solids holdup as the maximum quantity of solids that can be held in suspension in an agitated liquid. They present measurements of this factor for various values of gas velocity, gas distribution, solid-particle size, liquid surface tension, liquid viscosity, and a solid-liquid wettability parameter, and they propose the following two correlations in terms of dimensionless groups containing these parameters ... 

To construct a model which will give behavior similar to another bed, for example, a commercial bed, all of the dimensionless parameters listed in Eqs. (37) or (39) must have the same value for the two beds. The requirements of similar bed geometry is met by use of geometrically similar beds the ratio of all linear bed dimensions to a reference dimension such as the bed diameter must be the same for the model and the commercial bed. This includes the dimensions of the bed internals. The dimensions of elements external to the bed such as the particle return loop do not have to be matched as long as the return loop is designed to provide the proper external solids flow rate and size distribution and solid or gas flow fluctuations in the return loop do not influence the riser behavior (Rhodes and Laussman, 1992). 

Indices are dimensionless parameters derived from various mechanical and physical properties of the tablet blend and resulting compacts. Mechanical properties typically measured include indentation hardness (kinetic and static), elastic modulus, and tensile strength (10,11). Physical properties include particle size, shape, and size distribution, density (true, bulk, and tapped), flow properties and cohesive properties. 

The dimensionless retention parameter X of all FFF techniques, if operated on an absolute basis, is a function of the molecular characteristics of the compounds separated. These include the size of macromolecules and particles, molar mass, diffusion coefficient, thermal diffusion coefficient, electrophoretic mobility, electrical charge, and density (see Table 1, Sect. 1.4.1.) reflecting the wide variablity of the applicable forces [77]. For detailed theoretical descriptions see Sects. 1.4.1. and 2. For the majority of operation modes, X is influenced by the size of the retained macromolecules or particles, and FFF can be used to determine absolute particle sizes and their distributions. For an overview, the accessible quantities for the three main FFF techniques are given (for the analytical expressions see Table l,Sect. 1.4.1) ... 

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