Particle size equations

Particle-Size Equations It is common practice to plot size-distribution data in such a way that a straight line results, with all the advantages that follow from such a reduction. This can be done if the cui ve fits a standard law such as the normal probability law. According to the normal law, differences of equal amounts in excess or deficit from a mean value are equally likely. In order to maintain a symmetrical beU-shaped cui ve for the frequency distribution it is necessary to plot the population density (e.g., percentage per micron) against size. 

For several batches of the same catalyst with quite different mass median diameters, Gwyn found the exponent b to be constant, whereas the attrition constant Ka was found to decrease with mean particle size. Equation (1) is, therefore, valid for a particular size distribution only. Other... 

Particle Size. Equations 16, 21, and 8 with Eq. 24 or 25 show that the time needed to achieve the same fractional conversion for particles of different but unchanging sizes is given by... 

Many diffuse-reflectance instruments are available. Some employ several interference filters to provide narrow bands of radiation. Others are equipped with grating monochromators. Ordinarily, calibration is often a stringent requirement as samples must be acquired of the material for analysis that contain the range of analyte concentrations likely to be encountered. It may be useful to grind solid samples to a reproducible particle size. Equations are developed and used for the analysis. Once method development has been completed and validated, solid samples can be analyzed in a few minutes. Accuracy and precision are reported to be of 1 to 2% relative. 

Figure 10.23. Chamber dia. Dc from flow-rate, physical properties, and dso particle size (Equation 10.4,...

The frequency evaluated at one of the particle sizes equated to zero need not always exist For example, the Brownian aggregation frequency is unbounded when one of the particle sizes approaches zero. This is a reflection of the fact that aggregation is virtually certain between two particles when one of them is much smaller than the other and consequently capable of very rapid diffusional motion relative to the other much larger particle. In this case a(0, /y)/ cannot be expected to exist. 

Figure 5.12 Deviation from graphite data and Ni particle size (Equation 5.10). CHt decomposition, 500°C. Various nickel catalysts [378], [381]. Reproduced with the permission of Elsevier.

Subject to the assumption of infinitesimal particle size, the diffuse reflectance is a function only of the ratio of two constants, K and S, and not of their absolute values. Eor small particles (i.e., good approximations to infinitesimal particle size). Equation (3.38) can be used to quantitatively determine the concentration. If K is assumed to be proportional to the absorption coefficient obtained in transmission, the equation can be rewritten as shown in Equation (3.39), where u is the absorptivity of the analyte. 

The ratio has been shown to vary from about 90 for small particles to about 10 for large particles, which represents the theoretical limits of operation. In practice, these limits may be even narrower if elutriation is to be avoided because most practical fluidized beds contain a range of particle sizes. Equations (7.5.5)-(7.5.7) provide the theoretical limits of the superficial gas velocity at which elutriation can take place. Practical experience has shown, however, that in fluidized beds, the particles whose terminal falling velocity is smaller than the superficial gas velocity are not entrained (or elutriated) immediately but rather the elutriation occurs at a finite rate. 

Measurements of the diffusion arc thus usable for the determination of particle size. Equation (13) shows at the same time that D is inversely proportional to the radius of the particles it is therefore understandable that Graham considered colloid systems to be characterized by the low diffusion velocity of the dispersed substance. 

A combination of equation (C2.6.13), equation (C2.6.14), equation (C2.6.15), equation (C2.6.16), equation (C2.6.17), equation (C2.6.18) and equation (C2.6.19) tlien allows us to estimate how low the electrolyte concentration needs to be to provide kinetic stability for a desired lengtli of time. This tlieory successfully accounts for a number of observations on slowly aggregating systems, but two discrepancies are found (see, for instance, [33]). First, tire observed dependence of stability ratio on salt concentration tends to be much weaker tlian predicted. Second, tire variation of tire stability ratio witli particle size is not reproduced experimentally. Recently, however, it was reported that for model particles witli a low surface charge, where tire DL VO tlieory is expected to hold, tire aggregation kinetics do agree witli tire tlieoretical predictions (see [60], and references tlierein). 

If the magnitudes of the dissipative force, random noise, or the time step are too large, the modified velocity Verlet algorithm will not correctly integrate the equations of motion and thus give incorrect results. The values that are valid depend on the particle sizes being used. A system of reduced units can be defined in which these limits remain constant. 

Most tests of the validity of the BET area have been carried out with finely divided solids, where independent evaluation of the surface area can be made from optical microscopic or, more often, electron microscopic observations of particle size, provided the size distribution is fairly narrow. As already explained (Section 1.10) the specific surface obtained in this way is related to the mean projected diameter through the equation... 

Equation (2.28), being statistical in nature, requires a large number of particles to be measured, especially if the spread of particle size is wide. The possibility of error from this source is stressed by Arnell and Henneberry who found that in a particular sample of finely ground quartz, two particles in a total of 335 had a diameter about twenty times the most probable diameter, and that if these were overlooked the calculated value of A would be nearly doubled. 

It would be difficult to over-estimate the extent to which the BET method has contributed to the development of those branches of physical chemistry such as heterogeneous catalysis, adsorption or particle size estimation, which involve finely divided or porous solids in all of these fields the BET surface area is a household phrase. But it is perhaps the very breadth of its scope which has led to a somewhat uncritical application of the method as a kind of infallible yardstick, and to a lack of appreciation of the nature of its basic assumptions or of the circumstances under which it may, or may not, be expected to yield a reliable result. This is particularly true of those solids which contain very fine pores and give rise to Langmuir-type isotherms, for the BET procedure may then give quite erroneous values for the surface area. If the pores are rather larger—tens to hundreds of Angstroms in width—the pore size distribution may be calculated from the adsorption isotherm of a vapour with the aid of the Kelvin equation, and within recent years a number of detailed procedures for carrying out the calculation have been put forward but all too often the limitations on the validity of the results, and the difficulty of interpretation in terms of the actual solid, tend to be insufficiently stressed or even entirely overlooked. And in the time-honoured method for the estimation of surface area from measurements of adsorption from solution, the complications introduced by... 

Cyclone Efficiency. Most cyclone manufacturers provide grade-efficiency curves to predict overall collection efficiency of a dust stream in a particular cyclone. Many investigators have attempted to develop a generalized grade-efficiency curve for cyclones, eg, see (159). One problem is that a cyclone s efficiency is affected by its geometric design. Equation 15 was proposed to calculate the smallest particle size collectable in a cyclone with 100% efficiency (157). 

Table 13 can be used as a rough guide for scmbber collection in regard to minimum particle size collected at 85% efficiency. In some cases, a higher collection efficiency can be achieved on finer particles under a higher pressure drop. For many scmbbers the particle penetration can be represented by an exponential equation of the form (271—274)... 

The constant given the value 5 in equation 1 depends on particle size, shape, and porosity it can be assumed to be 5 for low porosities. Although equation 1 has been found to work reasonably well for incompressible cakes over narrow porosity ranges, its importance is limited in cake filtration because it cannot be used for most practical, compressible cakes. It can, however, be used to demonstrate the high sensitivity of the pressure drop to the cake porosity and to the specific surface of the soHds. 

B = outlet diameter or width, g = acceleration owing to gravity, m = 1 for circular opening and 0 for slotted opening, and 0 = hopper angle (measured from vertical) in degrees. A modification of this equation takes particle size into account. This modification is only important if the particle size is a significant fraction of the outlet size (8). 

The terminal velocity in the case of fine particles is approached so quickly that in practical engineering calculations the settling is taken as a constant velocity motion and the acceleration period is neglected. Equation 7 can also be appHed to nonspherical particles if the particle size x is the equivalent Stokes diameter as deterrnined by sedimentation or elutriation methods of particle-size measurement. 

Here again an equation is estabUshed (2) to describe the trajectory of a particle under the combined effect of the Hquid transport velocity acting in the x-direction and the centrifugal settling velocity in thejy-direction. Equation 13 determines the minimum particle size which originates from a position on the outer radius, and the midpoint of the space, between two adjacent disks, and just reaches the upper disk at the inner radius, r. Particles of this size initially located above the midpoint of space a are all collected on the underside of the upper disk those particles initially located below the midpoint escape capture. This condition defines the throughput, for which a 50% recovery of the entering particles is achieved. That is,... 

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