Disperse-phase number transport

The total number concentration N t, x) corresponds to the zeroth-order moment of the NDF (i.e. g = 1), and is defined by 

Note that the sign of the source term will depend on whether particles are created or destroyed in the system. Note also that the spatial transport term in Eq. (4.46) will generally not be closed unless, for example, all particles have identical velocities. The transport equation in Eq. (4.46) is mainly used for systems with particle aggregation and breakage (i.e. when N(t, x) is not constant). In such cases, it will typically be coupled to a system of moment-transport equations involving higher-order moments. 

---

The parameter p (= 7(5 ) in gas-liquid sy.stems plays the same role as V/Aex in catalytic reactions. This parameter amounts to 10-40 for a gas and liquid in film contact, and increases to lO -lO" for gas bubbles dispersed in a liquid. If the Hatta number (see section 5.4.3) is low (below I) this indicates a slow reaction, and high values of p (e.g. bubble columns) should be chosen. For instantaneous reactions Ha > 100, enhancement factor E = 10-50) a low p should be selected with a high degree of gas-phase turbulence. The sulphonation of aromatics with gaseous SO3 is an instantaneous reaction and is controlled by gas-phase mass transfer. In commercial thin-film sulphonators, the liquid reactant flows down as a thin film (low p) in contact with a highly turbulent gas stream (high ka). A thin-film reactor was chosen instead of a liquid droplet system due to the desire to remove heat generated in the liquid phase as a result of the exothermic reaction. Similar considerations are valid for liquid-liquid systems. Sometimes, practical considerations prevail over the decisions dictated from a transport-reaction analysis. Corrosive liquids should always be in the dispersed phase to reduce contact with the reactor walls. Hazardous liquids are usually dispensed to reduce their hold-up, i.e. their inventory inside the reactor. 

The rate constant k0 for orthokinetic coagulation is determined by physical parameters (velocity gradient du/dz, floe volume ratio of the dispersed phase, = sum over the product of particle number and volume), and the collision efficiency factor a0 observed under orthokinetic transport conditions ... 

In foams with charged gas/liquid interfaces, as in other disperse systems, various electrokinetic phenomena are possible to occur. Such are the change in the transport numbers of ions, electroosmosis, streaming potential and surface conductivity. While these phenomena are largely studied in disperse systems with solid disperse phase, the first electrokinetic observations in foams have been reported only recently. 

A number of other important potential applications of a micellar phase in supercritical fluids may utilize the unique properties of the supercritical fluid phase. For instance, polar catalyst or enzymes could be molecularly dispersed in a nonpolar gas phase via micelles, opening a new class of gas phase reactions. Because diffusivities of reactants or products are high in the supercritical fluid continuous phase, high transport rates to and from active sites in the catalyst-containing micelle may increase reaction rates for those reactions which are diffusion limited. 

The Microscopic Transport Equations for a Finite Number of Dispersed Phases - the Multi-Fluid Model... 

As in all mathematical descriptions of transport phenomena, the theory of polydisperse multiphase flows introduces a set of dimensionless numbers that are pertinent in describing the behavior of the flow. Depending on the complexity of the flow (e.g. variations in physical properties due to chemical reactions, collisions, etc.), the set of dimensionless numbers can be quite large. (Details on the physical models for momentum exchange are given in Chapter 5.) As will be described in detail in Chapter 4, a kinetic equation can be derived for the number-density function (NDF) of the velocity of the disperse phase n t, X, v). Also in this example, for clarity, we will assume that the problem has only one particle velocity component v and is one-dimensional in physical space with coordinate x at time t. Furthermore, we will assume that the NDF has been normalized (by multiplying it by the volume of a particle) such that the first three velocity moments are... 

The process of formulating mesoscale models from the microscale equations is widely used in transport phenomena (Ferziger Kaper, 1972). For example, heat transfer between the disperse phase and the fluid depends on the Nusselt number, and mass transfer depends on the Sherwood number. Correlations for how the Nusselt and Sherwood numbers depend on the mesoscale variables and the moments of the NDF (e.g. mean particle temperature and mean particle concentration) are available in the literature. As microscale simulations become more and more sophisticated, modified correlations that are based on the microscale results will become more and more common (Beetstra et al, 2007 Holloway et al, 2010 Tenneti et al, 2010). Note that, because the kinetic equation requires mesoscale models that are valid locally in phase space (i.e. for a particular set of mesoscale variables) as opposed to averaged correlations found from macroscale variables, direct numerical simulation of the microscale model is perhaps the only way to obtain the data necessary in order for such models to be thoroughly validated. For example, a macroscale model will depend on the average drag, which is denoted by... 

In summary, the Boussinesq-Basset, Brownian, and thermophoretic forces are rarely used in disperse multiphase flow simulations for different reasons. The Boussinesq-Basset force is neglected because it is needed only for rapidly accelerating particles and because its form makes its simulation difficult to implement. The Brownian and thermophoretic forces are important for very small particles, which usually implies that the particle Stokes number is near zero. For such particles, it is not necessary to solve transport equations for the disperse-phase momentum density. Instead, the Brownian and thermophoretic forces generate real-space diffusion terms in the particle-concentration transport equation (which is coupled to the fluid-phase momentum equation). 

To represent dispersion polymerization in conventional liquid media, several models have been reported in the literature, mainly focused on the particle formation and growth [33, 34] or on the reaction kinetics. Since our first aim is the reliable description of the reaction kinetics, we focus on the second type of models only. The model developed by Ahmed and Poehlein [35, 36], applied to the dispersion polymerization of styrene in ethanol, was probably the first one from which the polymerization rates in the two reaction loci have been calculated. A more comprehensive model was later reported by Saenz and Asua [37] for the dispersion copolymerization of styrene and butyl acrylate in ethanol-water medium. The particle growth as well as the entire MWD were predicted, once more evaluating the reaction rates in both phases and accounting for an irreversible radical mass transport from the continuous to the dispersed phase. Finally, a further model predicting conversion, particle number, and particle size distribution was proposed by Araujo and Pinto [38] for the dispersion polymerization of styrene in ethanol. 

In this modeling form, abbreviated as interacted liquid phase model, the liquid phase is considered as the system to be concerned aiming to obtain the transport information of the liquid phase. The dispersed phase is considered as the sm-roundings. The action of the dispersed phase (bubbles) on the liquid phase is treated as the external forces acting on the system (liquid phase). Thus, the evaluation of source term Su in Navier—Stokes equation of liquid phase should cover all the acting forces by the dispersed gas phase to the liquid phase. Such model can reduce the number of model equations and computer load. Computation shows that whether the interaction source term Su is properly considered, the final simulated result is substantially equal to that using two-fluid model (Fig. 3.7). 

---