Coagulation discrete

Fibrillated Fibers. Instead of extmding cellulose acetate into a continuous fiber, discrete, pulp-like agglomerates of fine, individual fibrils, called fibrets or fibrids, can be produced by rapid precipitation with an attenuating coagulation fluid. The individual fibers have diameters of 0.5 to 5.0 ]lni and lengths of 20 to 200 )Jm (Fig. 10). The surface area of the fibrillated fibers are about 20 m /g, about 60—80 times that of standard textile fibers. These materials are very hydrophilic an 85% moisture content has the appearance of a dry soHd (72). One appHcation is in a paper stmcture where their fine fiber size and branched stmcture allows mechanical entrapment of small particles. The fibers can also be loaded with particles to enhance some desired performance such as enhanced opacity for papers. When filled with metal particles it was suggested they be used as a radar screen in aerial warfare (73). 

Coagulation, i.e., the process by which discrete particles come in contact with each other in the air and remain joined together by surface forces, represents another way in which aerosol diameter will increase. However, it does not alter the mass of material in the coagulated particle. 

Exact solution of one-dimensional reversible coagulation reaction A+A A was presented in [108, 109] (see also Section 6.5). In these studies a dynamical phase transition of the second order was discovered, using both continuum and discrete formalisms. This shows that the relaxation time of particle concentrations on the equilibrium level depends on the initial concentration, if the system starts from the concentration smaller than some critical value, and is independent of the tia(0) otherwise. 

This case has been dealt with for continuous (rather than discrete) probability distributions in an earlier paper.27 Also, it is worth noting that the physics of this process are similar to the process of coagulation in atmospheric physics.37... 

The polymerization reaction takes place in the bulk and the formation of discrete particles is a result of the good solubility of the monomer in the solvent and the low solubility of formed particles of a certain size. Because this approach is dependent on the existence of such a kind of a solubility/non-solubility regime for the monomer, crosslinker, template, and polymer network, it might be hard to find the correct combination of compounds needed to interact in the prescribed way. Additionally, there is a high risk of coagulation with this method as no surfactant is added for the stabilization of the interface between the formed polymer and the monomer containing solution. 

Structure analysis of several proteases involved in blood coagulation and fibrinolysis reveals a diverse, sometimes repetitive, assembly of discrete protein modules (Fig. 9.4) [56]. While these modules represent independent structural units with individual folding pathways, their concerted action contributes to function and specificity in the final protein product. On the genetic level, these individual modules are encoded in separate exons. Over the course of modular protein evolution, new genes are created by duplication, deletion, and rearrangement of these exons. Mechanistically, the exon shuffling actually takes place in the intervening intron sequences (intronic recombination - for further details see [10]). 

Some results obtained by Storer (1968) using well-dialyzed polystyrene latices at pH 8.5 and mixtures of magnesium sulfate and sodium nitrate as the coagulating electrolytes are shown in Fig. 23. Distinct synergism was observed over the entire concentration range. The use of activities of the ions in the mixed system rather than concentrations, gave a reasonable explanation for the form of the data, bot it is also of interest that the discrete-ion treatment of Levine and Bell (1965) also predicts synergism for certain cases. 

BROWNIAN COAGULATION DYNAMICS OF DISCRETE DISTRIBUTION FOR AN INITIALLY MONODISPERSE AEROSOL... 

Browiiicm Coagulation Dynamics of Discrete Distribution for an Initially Monodisperse Aerosol 193... 

The solution for the discrete distribution (7.26) can be interpreted as the size distribution for the particles in a batch system at a time t after the start of coagulation. Alternatively, it is equivalent to the distribution after a residence time / in a plug flow system where / = jc/t/,. r is the distance from the entrance to the tube, and U is the average velocity. 

Substituting into (7.4). the equation of coagulation by laminar shear for the discrete spectrum becomes... 

A method of solving many coagulation anti agglomeration problems (Chapter 8) has been developed based on the use of a similarity transformation for the size distribution function (Swift and Fricdlander, 1964 Friedlander and Wang. 1966). Solutions found in this way are asymptotic forms approached after long times, and they are independent of the initial size distribution. Closed-form solutions for the upper and lower ends of the distribution can sometime.s be obtained in this way, and numerical methods can be used to match the solutions for intermediate-size particles. Alternatively, Monte Carlo and discrete sectional methods have been used to find solutions. 

The results of the numerical calculation are shown in Fig, 7.8, where they are compared with numerical calculations carried out for the discrete spectrum starting with an initially monodisperse system. There is good agreement between the two methods of calculation. Other calculations indicate that the similarity form is an asymptotic solution independent of the initial distributions so far studied. The values of a and b were found to be 0.9046 and 1.248, respectively. By (7,75) this corresponds to a 6.5% increase in the coagulation constant compared with the value for a monodisperse aerosol (7.21). The results of more recent calculations using a discrete sectional method are shown in Table 7.2. 

Figure 7.8 Sdt -prcscrving particle size distribution for Brownian coagulation, Tlie Ibnn is appaw-imatcly lognormal. The re.sult obtained by solution of the ordinary integrodiffereniial equation for the continuous spectrum is compared with the limiting solution of Hidy and Lilly (1965) for the discrete spectrum, calculated from the discrete form of the coagulation equation. Shown also are points calculated from analytical solutions for the lower and upper ends of the distribution (Friedlandcr and Wang. 1966).

Figure 7.12 Size distribution data of Fig. 7.) I for coagulation of small particles plotted in the coordinate. of the similarity theory. Shown also is the result of a Monte Carlo calculation for the discrete spectrum (Husar, 1971).

The change in the discrete distribution function with time and position is obtained by generalizing the equation of convective diffusion (Chapter 3) to include terms for particle growth and coagulation ... 

When a fast chemical reaction or a rapid quench leads to the formation of a high density of condensable molecules, panicle formation may take place either by homogeneous nucleation, an activated process, or by molecular "coagulation a process in which nearly all collisions are successful. What determines which of these processes controls In principle, this problem can be analyzed by solving the GDE for the discrete distribution discussed in the previous section. An approximate criterion proposed by Ulrich (1971) for determining whether nucleation or coagulation is the dominant process is based on the critical particle diameter d that appears in the theory of homogeneous nucleation (Chapter 9)... 

Brownian Coagulation Dynamics of Discrete Distribution for an Initially Monodisperse Aerosol 192 Brownian Coagulation Effect of Particle Force Fields 196 Effect of van der Waals Forces 197 Effect of Coulomb Forces 200 Collision Frequency for Laminar Shear 200 Simultaneous Laminar Shear and Brownian Motion 202 Turbulent Coagulation 204... 

In the previous sections, we focused our attention on calculation of the instantaneous coagulation rate between two monodisperse aerosol populations. In this section, we will develop the overall expression describing the evolution of a polydisperse coagulating aerosol population. A good place to start is with a discrete aerosol distribution. 

The Discrete Coagulation Equation A spatially homogeneous aerosol of uniform chemical composition can be fully characterized by the number densities of particles of various monomer contents as a function of time, Afc(f). The dynamic equation governing... 

Combining the coagulation production and depletion terms, we get the discrete coagulation equation ... 

For additional solutions of the discrete and continuous coagulation equations, the interested reader may wish to consult Drake (1972), Mulholland and Baum (1980), Tambour and Seinfeld (1980), and Pilinis and Seinfeld (1987). 

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