Mathematical models particle size distribution

Modeling the pore size in terms of a probability distribution function enables a mathematical description of the pore characteristics. The narrower the pore size distribution, the more likely the absoluteness of retention. The particle-size distribution represented by the rectangular block is the more securely retained, by sieve capture, the narrower the pore-size distribution. 

JR Crison, GL Amidon. The effect of particle size distribution on drug dissolution A mathematical model for predicting dissolution and absorption of suspensions in the small intestine. Pharm Res 10 S170, 1992. 

The extraction of more complex particle size distributions from PCS data (which is not part of the commonly performed particle size characterization of solid lipid nanoparticles) remains a challenging task, even though several corresponding mathematical models and software for commercial instruments are available. This type of analysis requires the user to have a high degree of experience and the data to have high statistical accuracy. In many cases, data obtained in routine measurements, as are often performed for particle size characterization, are not an adequate basis for a reliable particle size distribution analysis. 

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... 

The structure of the mathematical model is shown in Figure 1 where we have divided the model up into general balances, aqueous phase balances, individual particle balances, and particle size distribution balances - all of which exchange information with each other. To give an example of the form of the particle size distribution balances let us consider the total particle size distribution, F(V,t). 

A mathematical model, Allegra-Hawley [227,228], prediets the attenuation of ultrasonic waves as a funetion of frequency for each particle size distribution and concentration. Some mechanical, thermodynamic and transport properties of both phases are needed. The relationship between the size, concentration and frequency is obtained from the solution of the... 

Figure 30 shows a simple model of a typical laser diffraction instrument where the diffraction pattern of light scattered at various angles from the sample particles that pass through the He-Ne laser beam is measured by different detectors and recorded as numerical values relating to the scattering pattern. These numerical values are then converted to the particle size distribution in terms of the equivalent volume diameter using a mathematical model from the instrument s software. 

Simultaneous diffraction on more than one particle results in a superposition of the diffraction patterns of the individual particles, provided that particles are moving and diffraction between the particles is averaged out. This simplifies the evaluation, providing a parameter-free and model-independent mathematical algorithm for the inversion process (M. Heuer and K. Leschonski, Results Obtained with a New Instrument for the Measurement of Particle Size Distributions from Diffraction Patterns, Part. Part. Syst. Charact. 2, 7-13, 1985). 

Equation (21-66) estimates the variance of a random mixture, even if the components have different particle-size distributions. If the components have a small size (i.e., small mean particle mass) or a narrow particle-size distribution, that is, and c are low, the random mix s variance falls. Sommer has presented mathematical models for calculating the variance of random mixtures for particulate systems with a particle-size distribution (Karl Sommer, Sampling of Powders and Bulk Materials, Springer-Verlag, Berlin, 1986, p. 164). This model has been used for deriving Fig. 21-46. 

Hutchinson et al. [57] have examined the effects of particle size on the decomposition of lead azide powder. Finer powders (radius = 8 pm) decomposed more than twice as rapidly as coarser powders (radius = 24 pm). A mathematical model was developed which incorporated the independently measured particle-size distribution and gave an excellent description of the experimental sigmoid ar-time curves. 

Particle size distributions often produce denser packing structures because smaller particles may fill void spaces between larger ones. Based on this assumption several researchers have described densest packings. Table 6 shows porosities that can be obtained with binary or ternary particle mixtures (i.e. consisting of two or three particle sizes). Recently, a geometrical mathematical model for calculating the porosity of randomly packed binary mixtures of spherical particles has been developed. It could be shown that the absolute deviation between theoretical and experimentally obtained data... 

The mathematical model for the mass transfer of an adsorbate in the LC column packed with the silicalite crystal particles is based on the assumptions of (1) axial—dispersed plug—flow for the mobile phase with a constant interstitial flow velocity (2) Fickian diffusion in the silicalite crystal pore with an intracrystalline diffus— ivity independent of concentration and pressure and (3) spherical silicalite crystal particles with a uniform particle size distribution. A detailed discussion of these assumptions can be found in (13). The differential mass balances over an element of the LC column and silicalite crystal result in the following two partial differential equations ... 

Particle concentration and size distribution in raw water have extensive and complex effects on the performance of individual treatment units (flocculator, sedimentation tank, and filter) and on the overall performance of water treatment plants. Mathematical models of each treatment unit were developed to evaluate the effects of various raw water characteristics and design parameters on plant performance. The flocculation and sedimentation models allow wide particle size distributions to be considered. The filtration model is restricted to homogeneous suspensions but does permit evaluation of filter ripening. The flocculation model is formulated to include simultaneous flocculation by Brownian diffusion and fluid shear, and the sedimentation model is constructed to consider simultaneous contacts by Brownian diffusion and differential settling. The predictions of the model are consistent with results in water treatment practice. 

---