Particle size probability density

Information entropy offers a good possibility of becoming a consistent viewpoint to treat phenomena that must be expressed by the probability terms. By using information entropy, it will become possible to define the evaluation indices for mixing and separation operations/equipment, to estimate turbulent flow structure in a chemical equipment, to establish scale-up rules based on the turbulent flow structure, to present a general particle size probability density distribution, and to define the amount of anxiety/expectation. 

Particle size probability density distribution function (PSD function)... 

Daughter particle size probability density function (1/m [x]) Breakage probability function, determining the efficiency of the eddy-bubble collisions (—)... 

The reported minimum fluidization velocities for binary systems in the literature are probably the velocities of either (C/mf)M or (C/n,f)j. For binary systems of small particle size and density difference, these two minimum fluidization velocities may be taken to be similar. Some of the equations proposed for calculating the minimum fluidization velocity of a binary mixture are summarized below. 

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. 

Several colloidal systems, that are of practical importance, contain spherically symmetric particles the size of which changes continuously. Polydisperse fluid mixtures can be described by a continuous probability density of one or more particle attributes, such as particle size. Thus, they may be viewed as containing an infinite number of components. It has been several decades since the introduction of polydispersity as a model for molecular mixtures [73], but only recently has it received widespread attention [74-82]. Initially, work was concentrated on nearly monodisperse mixtures and the polydispersity was accounted for by the construction of perturbation expansions with a pure, monodispersive, component as the reference fluid [77,80]. Subsequently, Kofke and Glandt [79] have obtained the equation of state using a theory based on the distinction of particular species in a polydispersive mixture, not by their intermolecular potentials but by a specific form of the distribution of their chemical potentials. Quite recently, Lado [81,82] has generalized the usual OZ equation to the case of a polydispersive mixture. Recently, the latter theory has been also extended to the case of polydisperse quenched-annealed mixtures [83,84]. As this approach has not been reviewed previously, we shall consider it in some detail. 

Anderson (A2) has derived a formula relating the bubble-radius probability density function (B3) to the contact-time density function on the assumption that the bubble-rise velocity is independent of position. Bankoff (B3) has developed bubble-radius distribution functions that relate the contacttime density function to the radial and axial positions of bubbles as obtained from resistivity-probe measurements. Soo (S10) has recently considered a particle-size distribution function for solid particles in a free stream ... 

The need to separate solid and liquid phases is probably the most common phase separation requirement in the process industries, and a variety of techniques is used (Figure 10.9). Separation is effected by either the difference in density between the liquid and solids, using either gravity or centrifugal force, or, for filtration, depends on the particle size and shape. The most suitable technique to use will depend on the solids concentration and feed rate, as well as the size and nature of the solid particles. The range of application of various techniques and equipment, as a function of slurry concentration and particle size, is shown in Figure 10.10. 

The importance of chemical-reaction kinetics and the interaction of the latter with transport phenomena is the central theme of the contribution of Fox from Iowa State University. The chapter combines the clarity of a tutorial with the presentation of very recent results. Starting from simple chemistry and singlephase flow the reader is lead towards complex chemistry and two-phase flow. The issue of SGS modeling discussed already in Chapter 2 is now discussed with respect to the concentration fields. A detailed presentation of the joint Probability Density Function (PDF) method is given. The latter allows to account for the interaction between chemistry and physics. Results on impinging jet reactors are shown. When dealing with particulate systems a particle size distribution (PSD) and corresponding population balance equations are intro-... 

Figure 8.12 illustrates a solid particle impinging on a surface. It has been found that the erosive wear rate depends upon the impingement angle, a, the particle velocity, vq, and the size and density of the particle, as well as the properties of the surface material. It has also been found that there is a difference in erosive wear properties of brittle and ductile materials. The maximum erosive wear of ductile materials occurs at a = 20°, whereas the maximum erosive wear for brittle materials occurs near a = 90°. Since the impingement angle is probably lower than 90° for these type of flow situations, we might consider only brittle materials, such as ceramics for this application. Let us examine brittle erosive wear in a little more detail first. 

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