Particle number

Perdew J P, Parr R G, Levy M and Balduz J L Jr 1982 Density-functional theory for fractional particle number derivative discontinuities of the energy Phys. Rev. Lett. 49 1691-4... 

The Smith-Ewart expression (eq. 1) accurately predicts the particle number for hydrophobic monomers like styrene and butadiene (21), but fails to predict the particle number (22) for more hydrophilic monomers like methyl methacrylate and vinyl acetate. A new theory based on homogeneous particle... 

Stage II Growth in Polymer Particles Saturated With Monomer. Stage II begins once most of the micelles have been converted into polymer particles. At constant particle number the rate of polymerization, as given by Smith-Ewart kinetics is as follows (27) where is the... 

Using the same reasoning as with the particle number distribution above, we observe that if the x- and y-axes are provided with the nonlinear scales, n and tf, defined by Eqs. (14.34) and (14.35), the mass distribution m x)/m t) can be described by a straight line... 

Particle conservation in a vessel is governed by the particle-number continuity equation, essentially a population balance to identify particle numbers in each and every size range and account for any changes due to particle formation, growth and destruction, termed particle birth and death processes reflecting formation and loss of particulate entities, respectively. 

In the MSMPR crystallizer at steady state, the increase of particle number density brought about by particle growth and agglomeration is compensated by withdrawal of the product from the crystallizer. 

Film-crystal model concentration profiles of A, B and C and particle number density distributions are shown in Figure 8.14(a). 

At the crystallization stage, the rates of generation and growth of particles together with their residence times are all important for the formal accounting of particle numbers in each size range. Use of the mass and population balances facilitates calculation of the particle size distribution and its statistics i.e. mean particle size, etc. 

The kinetic mechanism of emulsion polymerization was developed by Smith and Ewart [10]. The quantitative treatment of this mechanism was made by using Har-kin s Micellar Theory [18,19]. By means of quantitative treatment, the researchers obtained an expression in which the particle number was expressed as a function of emulsifier concentration, initiation, and polymerization rates. This expression was derived for the systems including the monomers with low water solubility and partly solubilized within the micelles formed by emulsifiers having low critical micelle concentration (CMC) values [10]. 

The Smith-Ewart theory has been modified by several researchers [13,20-24]. These researchers argued against the Smith-Ewart theory that (1) the particle formation also occurs in the absence of micellar structure, (2) the predictions on particle number with the Smith-Ewart theory are higher relative to actual case. 

A minima] set of symmetric binary and triple collision rules conserving both momentum and particle number-... 

Consider a head-on collision between particles incoming along directions cp and Cfc+3. There are two possible outcomes such that both particle number and momentum are conserved the output must consist of two particles emerging either along directions c +i and 3 +4 (figure 9.10-b) or along Cfc i and (figure 9.10-c). We can either have the system always choose the same output channel, which... 

Whatever scheme we choose for treating head-on collisions, however, unless we also have triple collisions, spurious conservations laws are inevitable. In addition to the total particle numbei and momentum, it is easy to see that head-on collisions also conserve the difference in particle number in opposite directions-, that is, the difference in particle numbers in directions c and c +s- A simple way to fix this problem is to introduce a triple-collision of the form (cp, Ck+2, k+4) ( k+i, Ck+s, Ck+5)... 

It is easy to invent rules that conserve particle number, energy, momentum and so on, and to smooth out the apparent lack of structural symmetry (although we have cheated a little in our example of a random walk because the circular symmetry in this case is really a statistical phenomenon and not a reflection of the individual particle motion). The more interesting question is whether relativistically correct (i.e. Lorentz invariant) behavior can also be made to emerge on a Cartesian lattice. Toffoli ([toff89], [toffSOb]) showed that this is possible. 

If only one type of particle is present, mx = m2 however, the expressions relating the velocities before and after collision do not simplify to any great extent. If several types of particles are present, then there results one Boltzmann equation for the distribution function for each type of particle in each equation, integrals will appear for collisions with each type of particle. That is, if there are P types of particles, numbered i = 1,2,- , P, there are P distribution functions, ft /(r,vt, ), describing the system ftdrdvt is the number of particles of type i in the differential phase space volume around (r,v(). The set of Boltzmann equations for the system would then be ... 

It represents the probability of finding the particles numbered from 1 to in the given configuration, without regard to the configuration of the particles numbered from n + 1 to N. In particular ... 

Insol particles, number of particles retained on No 60 US standard sieve... 

Type penetration Speed Particle number Charge Example... 

EPM has been developed to simulate as a function of time all the phases, species, and the detailed )tinetic mechanism of the previous section. The structure of EPM consists of material balances, the particle number concentration balance, an energy balance, and the calculation of important secondary variables. 

Rate of Reactions. The rates of reaction in the aqueous and polymer phases were calculated using the appropriate kinetic constants according to the kinetic mechanisms described above, radical and molecular concentrations, and the particle number concentration. 

A dynamic ordinary differential equation was written for the number concentration of particles in the reactor. In the development of EPM, we have assumed that the size dependence of the coagulation rate coefficients can be ignored above a certain maximum size, which should be chosen sufficiently large so as not to affect the final result. If the particle size distribution is desired, the particle number balance would have to be a partial differential equation in volume and time as shown by other investigators ( ). 

There is an extensive amount of data in the literature on the effect of many factors (e.g. temperature, monomer and surfactant concentration and types, ionic strength, reactor configuration) on the time evolution of quantities such as conversions, particle number and size, molecular weight, composition. In this section, EPM predictions are compared with the following limited but useful cross section of isothermal experimental data ... 

Figure 2. Particle number vs. ionic strength for the styrene data of Goodwin et al. (22.) ...

Figure 6. Particle number vs. soap concentration for the methyl methacrylate data of Siitterlin (2ji). ...

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