Particulate phase equations

The scavenging model described in Equation (4) is robust and has been applied in a number of instances (e.g., Kaufman et al. 1981). However, a more detailed description of Th scavenging results from a model that treats the dissolved and particulate phases separately (e.g., Krishnaswami et al. 1976 Fig. 1). For dissolved " Th ... 

The equations of motion can either be formulated for individual particles and the surrounding fluid, or the fluid and the particulate phases can each be considered a continuum. Both approaches yield identical results, see Glicksman et al. (1994) for a complete derivation. For our purposes, we will base the derivation on the continuum model formulated by Jackson. 

In this Eq. (Js)n is the Jacobi matrix for the solid phase, which contains the derivatives of the mass residuals for the particulate phase to the solid volume fraction. Explicit expressions for the elements of the Jacobi matrix can be obtained from the continuity for the solid phase and the momentum equations. For example for the central element, the following expression is obtained from the solid phase continuity equation, in which the convective terms are evaluated with central finite difference expressions ... 

Though this new algorithm still requires some time step refinement for computations with highly inelastic particles, it turns out that most computations can be carried out with acceptable time steps of 10 5 s or larger. An alternative numerical method that is also based on the compressibility of the dispersed particulate phase is presented by Laux (1998). In this so-called compressible disperse-phase method the shear stresses in the momentum equations are implicitly taken into account, which further enhances the stability of the code in the quasi-static state near minimum fluidization, especially when frictional shear is taken into account. In theory, the stability of the numerical solution method can be further enhanced by fully implicit discretization and simultaneous solution of all governing equations. This latter is however not expected to result in faster solution of the TFM equations since the numerical efforts per time step increase. 

The vertical scavenging model also allows one to predict the distribution of particulate radionuclide profiles. Following Craig et al. [53] the particulate phase activity would be given by the solution of the equation ... 

Although Rs values of high Ks compounds derived from Eq. 3.68 may have been partly influenced by particle sampling, it is unlikely that the equation can accurately predict the summed vapor plus particulate phase concentrations, because transport rates through the boundary layer and through the membrane are different for the vapor-phase fraction and the particle-bound fraction, due to differences in effective diffusion coefficients between molecules and small particles. In addition, it will be difficult to define universally applicable calibration curves for the sampling rate of total (particle -I- vapor) atmospheric contaminants. At this stage of development, results obtained with SPMDs for particle-associated compounds provides valuable information on source identification and temporal... 

Substitution of eq. (5.396) into the material balance for the particulate phase (eq. (3.518)) gives an equation for C, which can be solved analytically for certain reaction kinetics. Finally, the reactant concentration at the exit of the bed C0 is... 

When a larger fraction of HOC was associated with the particulate phase, numerical solutions of Equation (6.132) and Equation (6.133) were required to predict PCB transformation rates. The instantaneous concentration of dissolved PCB was estimated by incorporating the terms S and OH into the rate constant expressions for Equation (6.133) and Equation (6.134). 

The particulate phase in the annular zone of a spouted bed can be described as an isotropic, incompressible, rigid plastic, non-cohesive Coulomb powder. Assuming that this material is in a quasi-static critical condition, the stress field can be described by equations developed for a static material element. 

This chapter describes the fundamental principles of heat and mass transfer in gas-solid flows. For most gas-solid flow situations, the temperature inside the solid particle can be approximated to be uniform. The theoretical basis and relevant restrictions of this approximation are briefly presented. The conductive heat transfer due to an elastic collision is introduced. A simple convective heat transfer model, based on the pseudocontinuum assumption for the gas-solid mixture, as well as the limitations of the model applications are discussed. The chapter also describes heat transfer due to radiation of the particulate phase. Specifically, thermal radiation from a single particle, radiation from a particle cloud with multiple scattering effects, and the basic governing equation for general multiparticle radiations are discussed. The discussion of gas phase radiation is, however, excluded because of its complexity, as it is affected by the type of gas components, concentrations, and gas temperatures. Interested readers may refer to Ozisik (1973) for the absorption (or emission) of radiation by gases. The last part of this chapter presents the fundamental principles of mass transfer in gas-solid flows. 

Deposition of PCBs from the atmosphere includes wet and dry depositional processes. Estimates of these processes can be obtained from measured air concentrations in the vapour and particulate phases which are applied to mass transfer equations, or they can be measured directly. 

The available continuum models for dispersed multi-phase flows thus follow one of two asymptotic approaches. The dilute phase approach is formulated based on the continuum mechanical principles in terms of the local conservation equations for each of the phases. A macroscopic model is then obtained by averaging the local equations based on an appropriate averaging procedure. In the dense phase approach, on the other hand, a kinetic theory description is adopted for the dispersed particulate phase (granular material), whereas an averaged continuum model formulation is adopted for the interstitial phase. 

In most gas-particle flow situations occurring in fluidized bed reactors, a standard k — e turbulence model is used to describe the turbulence in the continuous phase whereas a separate transport equation is formulated for the kinetic energy (or granular temperature) of the particulate phase [122, 42, 41, 165, 84, 52]. Further details on granular flows are given in chap 4. 

The PT model represents an extension of the basic CPV model and contains extended closures for the particle collisional pressure and the particle-particle velocity correlation terms, as well as simple attempts to account for some of the gas-particle interaction phenomena. For the gas phase, on the other hand, the same set of transport equations as for the CPV model are employed. The particulate phase continuity equation is also the same, but the momentum equation for the particulate phase is modified. 

To model the particle velocity fluctuation covariances caused by particle-particle collisions and particle interactions with the interstitial gas phase, the concept of kinetic theory of granular flows is adapted (see chap 4). This theory is based on an analogy between the particles and the molecules of dense gases. The particulate phase is thus represented as a population of identical, smooth and inelastic spheres. In order to predict the form of the transport equations for a granular material the classical framework from the kinetic theory of... 

The transport equation for the granular temperature 9p, written in terms of the turbulent kinetic energy analogue of the particulate phase kp, is given by [64, 65, 109] ... 

For the particulate phase, the PT-model equations that were described in sect 10.7.4 are used with minor extensions. That is, in the PGT-model... 

The effective particle phase viscosity is still obtained from (10.123). In addition, the turbulent viscosity of the particulate phase is calculated from (10.125) in which kgp is obtained from a separate balance equation. The interaction time between the particle motion and the gas velocity fluctuations Tgp, is modeled as suggested by Csanady [25] ... 

Proper boundary conditions are generally required for the primary variables like the gas and particle velocities, pressures and volume fractions at all the vessel boundaries as these model equations are elliptic. Moreover, boundary conditions for the granular temperature of the particulate phase is required for the PT, PGT and PGTDV models. For the models including gas phase turbulence, i.e., PGT and PGTDV, additional boundary conditions for the turbulent kinetic energy of the gas phase, as well as the dissipation rate of the gas phase and the gas-particle fluctuation covariance are required. The... 

Equation 10.9 for a particulate phase that was a solid) that is closer to that described by the unsteady-state convective mass transfer mechanism. 

The continuous phase variables, which affect the behavior of each particle, may be collated into a finite c-dimensional vector field. We thus define a continuous phase vector Y(r, t) = [7 (r, t), 2(1, t. .., l (r, t)], which is clearly a function only of the external coordinates r and time t. The evolution of this field in space and time is governed by the laws of transport and interaction with the particles. The actual governing equations must involve the number density of particles in the particulate phase, which must first be identified. 

Referring to Equation (16.2), we therefore need to assess the effect of an increasing mean particle size (i.e. from a mean size of 130 pm to 350 pm) on ep (average particulate phase voidage) and es (bubble fraction)... 

As a first approximation, we will the simple two phase theory of fluidization (see Chapter 7), which means that the average particulate phase voidage will be taken as the voidage at minimum fluidization f and the bubble fraction will be given by Equation (7.28) ... 

The particulate phase is an incompressible fluid with a bulk density similar to that of a fluidized bed at minimum fluidization. The continuity equation of the particles can thus be expressed as... 

In this case, the fluidizing fluid moves upward relative to the bubble motion. This case is usual for beds of large particles and small bubbles. Since R is less than zero in this case, the majority of fluidizing fluid enters the void at the base and leaves from the roof. The fluidizing fluid, in essence, uses the bubble void as the shortcut. The fluidizing fluid penetrates the particulate phase freely from the bubble except for a small fraction of fluid in the shaded area, as shown in Fig. 12 in a circle of radius a, expressed in the following equation. 

It was assumed in the Ostergaard model that the bed consists of a liquid particulate phase a bubble phase and a wake phase. It was considered as well that the wake moves with the bubble velocity and their porosity was identical to that of the liquid fluidized phase. Under these assumptions, the following equation was... 

With these two equations, an expression of the Richardson--Zaki type applied to the particulate phase and a correlation function between Sq and Uj an iterative method of calculation of e was proposed. The (gq + e ) results predicted with this method and the experimental values differed significantly,... 

For the particulate phase one can write the equations of motion with time-averaging procedures as... 

Combine the equation of motion for the fluid and particulate phase in the 6 direction and integrate the result to obtain an expression for vV in terms of the particle velocity fluctuations. 3-11 Interpret Fig. 4-13 in light of the expressions derived in Eqs. 3-82 to 3-85. 

The suspended sediment/water partition coefficients have been measured for 19 chlorinated organics in 25 samples from the St. Clair, Detroit and Niagara Rivers. An excellent linear correlation (r = 0.87) between the organic-carbon corrected partition coefficient (Kqq) and the octanol/water partition coefficient (Kq, ) was found (log = 0.76 log + 1.66). Using this equation plus another equation developed in this paper it is shown that the percentage of chemical in the dissolved and particulate phases in the study rivers could be estimated from a chemical s to within a factor of two. The paper also discusses the time required for equilibrium to be achieved between the dissolved and particulate phases, and the potential importance of biota such as algae in the partitioning process. 

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