Particle number concentration balance

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. 

In 1987 Valdes104 developed a model for composite deposition at a RDE taking into account the various ways in which a particle is transported to the cathode surface. As starting point an equation of continuity for the particle number concentration, C p, based on a differential mass balance was chosen, that is ... 

One of the main features of concentrated suspensions is the formation of three-dimensional structure units, which determine their properties and in particular their rheology. The formation of these units is determined by the interparticle interactions, which need to be clearly defined and quantified. It is useful to define the concentration range above which a suspension may be considered concentrated. The particle number concentration and volume fraction, tj), above which a suspension may be considered concentrated is best defined in terms of the balance between the particle translational motion and interparticle interaction. At one ex-... 

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 ( ). 

All quantitative theories based on micellar nucleation can be developed from balances of the number concentrations of particles, and of the concentrations of aqueous radicals. Smith and Ewart solved these balances for two limiting cases (i) all free radials generated in the aqueous phase assumed to be absorbed by surfactant micelles, and (ii) micelles and existing particles competing for aqueous phase radicals. In both cases, the number of particles at the end of Interval I in a batch macroemulsion polymerization is predicted to be proportional to the aqueous phase radical flux to the power of 0.4, and to the initial surfactant concentration to the power of 0.6. The Smith Ewart model predicts particle numbers accurately for styrene and other water-insoluble monomers. Deviations from the SE theory occur when there are substantial amounts of radical desorption, aqueous phase termination, or when the calculation of absorbance efficiency is in error. 

Hulburt and Katz (HI7) developed a framework for the analysis of particulate systems with the population balance equation for a multivariate particle number density. This number density is defined over phase space which is characterized by a vector of the least number of independent coordinates attached to a particle distribution that allow complete description of the properties of the distribution. Phase space is composed of three external particle coordinates x and m internal particle coordinates Xj. The former (Xei, x 2, A es) refer to the spatial distribution of particles. The latter coordinate properties Ocu,Xa,. . , Xt ) give a quantitative description of the state of an individual particle, such as its mass, concentration, temperature, age, etc. In the case of a homogeneous dispersion such as in a well-mixed vessel the external coordinates are unnecessary whereas for a nonideal stirred vessel or tubular configuration they may be needed. Thus (x t)d represents the number of particles per unit volume of dispersion at time t in the incremental range x, x -I- d, where x represents both coordinate sets. The number density continuity equation in particle phase space is shown to be (HI 8, R6)... 

The population balance equation is employed to describe the temporal and steady-state behavior of the droplet size distribution for physically equilibrated liquid-liquid dispersions undergoing breakage and/or coalescence. These analyses also permit evaluation of the various proposed coalescence and breakage functions described in Sections III,B and C. When the dispersion is spatially homogeneous it becomes convenient to describe particle interaction on a total number basis as opposed to number concentration. To be consistent with the notation employed by previous investigators, the number concentration is replaced as n i,t)d i = NA( i t)dXi, where N is the total number of particles per unit volume of the dispersion, and A(xj t) dXi is the fraction of drops in increment X, to X( + dxi- For spatially homogeneous dispersions such as in a well-mixed vessel, continuous flow of dispersions, no density changes, and isothermal conditions Eq. (102) becomes... 

The population balance model of Mohler et al. [4], which forms the basis for the presented approach, describes the replication of influenza A viruses in MDCK cells growing on microcarriers. It is rmstmctured and includes three concentrated state variables, which are the number concentrations of uninfected cells Uc, infected cells /<, and free virus particles V. 

Extensions of the early model employed mass balances across all components.i The model agreed well with experimental data for the production of polystyrene. However, as in most traditional models it was a particle number model, based on all particles being identical. These models helped analyze polymerization in terms of the number concentration of polymer particles containing actively polymerizing radicals of number, Ai(t). As a result, they contained no particle size... 

In a simple model, when the considered volume is filled with identical particles engaged in the process of Brownian coagulation, and the suspension is assumed to remain monodisperse, the change of number concentration of particles with time is described by the balance equation for the particles ... 

This is the population balance equation (PBE). Its principal form was first given by Smoluchowski (1916). In Eq. (4.1), denotes the aggregate number concentration, while Kjj is a kinetic constant related with the collision between clusters of mass i and j, and Etj is the sticking probability of the sub-clusters. Commonly, it is assumed that any successful collision (i.e. surface contact) leads to a stable particle bond and that Ey can be, thus, set to 1. 

To construct an overall rate law from a mechanism, write the rate law for each of the elementary reactions that have been proposed then combine them into an overall rate law. First, it is important to realize that the chemical equation for an elementary reaction is different from the balanced chemical equation for the overall reaction. The overall chemical equation gives the overall stoichiometry of the reaction, but tells us nothing about how the reaction occurs and so we must find the rate law experimentally. In contrast, an elementary step shows explicitly which particles and how many of each we propose come together in that step of the reaction. Because the elementary reaction shows how the reaction occurs, the rate of that step depends on the concentrations of those particles. Therefore, we can write the rate law for an elementary reaction (but not for the overall reaction) from its chemical equation, with each exponent in the rate law being the same as the number of particles of a given type participating in the reaction, as summarized in Table 13.3. 

Routh and Russel [10] proposed a dimensionless Peclet number to gauge the balance between the two dominant processes controlling the uniformity of drying of a colloidal dispersion layer evaporation of solvent from the air interface, which serves to concentrate particles at the surface, and particle diffusion which serves to equilibrate the concentration across the depth of the layer. The Peclet number, Pe is defined for a film of initial thickness H with an evaporation rate E (units of velocity) as HE/D0, where D0 = kBT/6jT ir- the Stokes-Einstein diffusion coefficient for the particles in the colloid. Here, r is the particle radius, p is the viscosity of the continuous phase, T is the absolute temperature and kB is the Boltzmann constant. When Pe 1, evaporation dominates and particles concentrate near the surface and a skin forms, Figure 2.3.5, lower left. Conversely, when Pe l, diffusion dominates and a more uniform distribution of particles is expected, Figure 2.3.5, upper left. 

The existence of radial temperature and concentration gradients means that, in principle, one should include terms to account for these effects in the mass and heat conservation equations describing the reactor. Furthermore, there should be a distinction between the porous catalyst particles (within which reaction occurs) and the bulk gas phase. Thus, conservation equations should be written for the catalyst particles as well as for the bulk gas phase and coupled by boundary condition statements to the effect that the mass and heat fluxes at the periphery of particles are balanced by mass and heat transfer between catalyst particles and bulk gas phase. A full account of these principles may be found in a number of texts [23, 35, 36]. 

At one extreme diffusivity may be so low that chemical reaction takes place only at suface active sites. In that case p is equal to the fraction of active sites on the surface of the catalyst. Such a polymer-supported phase transfer catalyst would have extremely low activity. At the other extreme when diffusion is much faster than chemical reaction p = 1. In that case the observed reaction rate equals the intrinsic reaction rate. Between the extremes a combination of intraparticle diffusion rates and intrinsic rates controls the observed reaction rates as shown in Fig. 2, which profiles the reactant concentration as a function of distance from the center of a spherical catalyst particle located at the right axis, When both diffusion and intrinsic reactivity control overall reaction rates, there is a gradient of reactant concentration from CAS at the surface, to a lower concentration at the center of the particle. The reactant is consumed as it diffuses into the particle. With diffusional limitations the active sites nearest the surface have the highest turnover numbers. The overall process of simultaneous diffusion and chemical reaction in a spherical particle has been described mathematically for the cases of ion exchange catalysis,63 65) and catalysis by enzymes immobilized in gels 66-67). Many experimental parameters influence the balance between intraparticle diffusional and intrinsic reactivity control of reaction rates with polymer-supported phase transfer catalysts, as shown in Fig. 1. 

In the bottom-up approach, a large variety of ordered nano-, micro-and macrostructures may be obtained by changing the balance of all the attractive and repulsive forces between the structure-forming molecules or particles. This can be achieved by altering the environmental conditions (temperature, pH, ionic strength, presence of specific substances or ions) and the concentration of molecules/particles in the system (Min et al., 2008). As this takes place, the interrelated processes of formation and stabilization are both important considerations in the production of nanoparticles. In addition, as particles grow in size a number of intrinsic properties change, some qualitatively, others quantitatively some affect the equilibrium (thermodynamic) properties, and others affect the nonequilibrium (dynamic) properties such as relaxation times. 

Reaction only takes place in the dense phase since that is where the catalyst particles are. Since the exchange is with a uniform environment where the concentration is cp, we can see that by the time the bubble has reached the top of the bed, the concentration of reactant in it is cp + (co - cp).exp -Tr, where c0 is the entering concentration, H the height of the bed and Tr = QH/UaV is a dimensionless transfer number. By doing a mass balance on the dense phase as a whole14 we obtain a linear equation for cp in terms of the... 

Nomura and Harada already reported an experimental and theoretical study on the effect of lowering the amount of monomer initially charged on the number of polymer particles formed in a batch reactor(14). Under usual conditions in batch operation, micelles disappear and the formation of particles terminates before the disappearance of monomer droplets in the water phase. However, if the initial monomer concentration is extremely low, micelles would exist even after the disappearance of monomer droplets and hence, particle formation will continue until all emulsifier molecules are adsorbed on the surfaces of polymer particles. This condition is quantitatively expressed by the following emulsifier balance equation., ... 

---