Particle Balance

For a steady-state ciystallizer receiving sohds-free feed and containing a well-mixed suspension of ciystals experiencing neghgible breakage, a material-balance statement degenerates to a particle balance (the Randolph-Larson general-population balance) in turn, it simplifies to... 

When a spherical particle of diameter d settles in a viscous liquid under earth gravity g, the terminal velocity V, is determined by the weight of the particle-balancing buoyancy and the viscous drag on the... 

Polymer Particle Balances (PEEK In the case of multiconponent emulsion polymerization, a multivariate distribution of pjarticle propierties in terms of multiple internal coordinates is required in this work, the polymer volume in the piarticle, v (continuous coordinate), and the number of active chains of any type, ni,n2,. .,r n (discrete coordinates), are considered. Therefore... 

In order to be able to explain the observed results plasma modeling was applied. A one-dimensional fluid model was used, which solves the particle balances for both the charged and neutral species, using the drift-diffusion approximation for the particle fluxes, the Poisson equation for the electric field, and the energy balance for the electrons [191] (see also Section 1.4.1). 

Under conditions where a spherical particle is not completely supported by the forces attributable to the yield stress, it will settle at a velocity such that the total force exerted by the fluid on the particle balances its weight. 

As the net velocity of the particle is increased, the viscous force Fv opposing its motion also increases. Soon this force, shown in Figure 2.2b, equals the net driving force responsible for the motion. Once the forces acting on the particle balance, the particle experiences no further acceleration and a stationary state velocity is reached. It may be shown that, under stationary state conditions and for small velocities, the force of resistance is proportional to the stationary state velocity v ... 

For a fluidized standpipe, the drag force of particles balances the pressure head as a result of the weight of solids. If the Richardson and Zaki form of equation (Eq. (8.55)) is proposed for the drag force, derive an expression for the leakage flow of gas in this standpipe. Discuss the effect of particle size on the leakage flow assuming all other conditions are maintained constant. 

Here S, are elements of the matrix kernel M. The latter is similar to that in Eq. (3.470) but of a higher rank (6x6). The particle balance provides all other equations ... 

To close the set (3.544), one has to take into account the particle balance... 

Settling of a particle with radius a in a dilute suspension is hindered by the drag exerted in a dispersion medium. The resistance of the medium is proportional to the settling velocity of the particle. In a very short time, the particle reaches a constant velocity, known as the terminal velocity. The gravitational force on the particle balances the hydrodynamic resistance of the medium as 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). 

Notice how the catalyst particle balances are coupled to the reactor mass and energy balances. Thus, the catalyst particle balances [Equation (10.2.7)] must be solved at each position along the axial reactor dimension when computing the bulk mass and energy balances [Equation (10.2.6)]. Obviously, these solutions are lengthy. 

An example of the use of the population balance method to predict reaction in particulate systems is presented in the work of Min and Ray (M16, M17). The authors developed a computational algorithm for a batch emulsion polymerization reactor. The model combines general balances, individual particle balances, and particle size distribution balances. The individual particle balances were formulated using the population balance... 

Maintain particle balance—create new growth centers in the absence of nucleation. 

A model for the kinetics of aggregation and sedimentation in lakes has been presented elsewhere (O Melia, 1980) a short summary is given here. The approach begins with a particle balance for the epilimnion of a lake ... 

The total mass of crystals per volume of liquid is same in both crystallizer and product magmas. For a volume V of product, by a particle balance, the number of crystals in the product equals the number of crystals per unit time. The mass of a single product crystal is... 

Figure 7.25 Fixed-bed reactor volume element containing fluid and catalyst particles the equations show the coupling between the catalyst particle balances and the overall reactor balances.

For a bed of uniform spherical collectors oriented perpendicular to the uniform flow velocity upstream of the medium 17, we can write an expression for the average suspended particle balance over a differential slice of bed of depth dx (Spielman FitzPatrick 1973). The equation for the differential rate... 

Valuable information about the physics involved in the kinetic treatment of a specific problem can be obtained by considering the consistent macroscopic balance equations of the electrons, Eqs. (31) to (33), adapted to the specific kinetic problem. On the right side of the power and momentum balance, Eqs. (32) and (33), a difference between the corresponding gain fi-om the electric field and the total loss in collisions occurs. Gain and loss terms arise on the right side of the particle balance equation, Eq. (31), too if nonconservative electron collision processes (for instance, ionization and attachment) are additionally taken into account in the kinetic equation, Eq. (8), and thus in the equation system (12). 

In this case, the electron particle balance [Eq. (31)] is automatically satisfied. This means that for plasmas in steady state, no restriction on the electron density exists and the density can be freely chosen. 

It should be mentioned that the additional inclusion of nonconservative electron collision processes in the kinetic study of a plasma in steady state does not really make sense from a strict point of view. In such a case, fulfillment of the consistent electron particle balance would require the production and the loss of electrons to completely compensate for each other in any small volume of the plasma and at any time. But, for a given gas and its specific atomic data, such a requirement can not naturally be satisfied for any reduced field strength. Thus, E/N would no longer be a parameter of Eq. (36) if it was extended to nonconservative collision processes. 

Since, again, in time-dependent conditions, only conservative collisions are considered for simplicity, the consistent particle balance, Eq. (45), simply says that the electron density n is time-independent. Because of the linear dependence of all terms of the basic system (44) on the isotropic or anisotropic distribution, instead of the latter, the one-electron normalized distributions /o(L/, t)/n and fy U,t)/n can immediately be introduced into this system. If nonconservative collisions are also considered, the electron density n t) becomes time-dependent, and such a normalization will no longer be possible. 

By using the same approach and an adequate choice of the boundary condition, the spatial evolution of the electron kinetic quantities can be analyzed in all those space-dependent electric fields that do not reverse their direction with growing z. In such studies, nonconservative inelastic electron collisions can also be included and will cause, in accordance with the consistent particle balance, Eq. (56), a spatial evolution of the particle current density y (z) also. 

Consider a particle of mass m falling in a gravity field through a viscous medium. The gravitational force is opposed by the inertial force, ma = m dv/dt and the frictional force, which is proportional to the velocity of the particle. Balancing these forces we have... 

Let Ma, Afe, Mq be the mass concentrations of species A in the gas phase and of species B(s) and C(s) in the aerosol phase, respectively. Let 5a(0 and 5b(/) be the amounts of A(g) and B(s) that have been deposited on the ground up to the time t. For simplicity it is assumed that the aerosol particles are monodisperse (diameter Dp), Me Mb at all times, and that there is a source of C(s) particles balancing their deposition so that Me and the number of particles can be considered constant with time. As a result of these assumptions the particle diameter is not influenced by the conden-.sation/evaporation of A(g) and is assumed to be constant with time. 

In correspondence ivith the guidelines given above, the catalyst effectiveness is mostly only a weak function of p, and y. Therefore p >= //( ). The latter function is available in analytical form for first-order reactions [38]. In the general case, it has to be determined from the numerical solution of the catalyst particle balances, as done for instance in [35, 39-42]. In the case of negative reaction orders, multiple effectiveness factors can be obtained [32]. This is illustrated in Fig. 5.27 for the MTBE synthesis where the rate has a negative order with respect to the educt methanol. 

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