## VU dP V n

Energy exchange between the fluid and the solid phase is modeled as

In order to fully understand the physical significance of various terms appearing in the governing equations of granular flows, the references cited above can be consulted. Detailed modeling of the various terms appearing in the granular flow equations is a fast developing field. It must be mentioned here that the modeler has to be careful to ensure consistency in all the different formulations when finalizing the complete set of governing equations. This is especially true when there are more than one solid phases (see Mathiesen (1997) and Mathiesen et al. (2000) for more discussion about the consistency in model formulations).

Other conservation equations (enthalpy and species) for multiphase flows can be written following a similar general format. For example, the enthalpy conservation equation is written:

— (akpkhk)+V ■ (akPkUkhk) = -V ■ q+(tk : VUk)+Sk ——+ > (Qpk + Spkhpk)+Sk d t Dt p=1

where hk is the specific enthalpy of phase k and Sk is the source of enthalpy (for example, due to chemical reaction). Qpk is the energy transfer between the pth and kth phase. Spkhpk is the energy transfer associated with the mass transfer between p and k phases. Heat or mass transfer between phases must satisfy the local balance condition:

The species conservation equation for multiphase flows can be written:

d n — (akpkmik) + V ■ (akpUmk ) = V ■ (akpkDk Vm!k) + J^ (Spk ) (4.49)

where m'k is the mass fraction of species i in phase k. Spp (mass transfer from phase p to phase p) is zero. The rate of energy or mass transfer between phases can be written:

where the second term on the right-hand side represents the temperature or concentration driving force and the first term represents heat or mass transfer coefficient. An appropriate value of heat or mass transfer coefficient can be obtained by using suitable correlations of Nusselt number (in terms of Reynolds number and Prandtl number) or Sherwood number (in terms of Reynolds number and Schmidt number). Some typical correlations to estimate fluid-particle heat transfer coefficients and mass transfer coefficients are listed in Appendix 4.3. Evaporation of volatile fluid from dispersed phase particles needs special attention. Several different models with varying degrees of sophistication have been proposed. More often than not, semi-empirical models based on saturation and boiling temperatures along with time relaxation parameters are adequate to simulate evaporation processes in chemical reactors. In these models, source due to evaporation or condensation is generally expressed as rvalPl(Tl- rsat ) mm v =----(4.51)

T sat where Tsat is a saturation temperature and rv is an empirical parameter controlling the rate of evaporation. See, for example, Theologos et al. (1997) on modeling the evaporation of liquid oil in the bottom section of a FCC riser reactor.

Boundary conditions for Eulerian multiphase flow models can be formulated using the usual practices discussed in Chapter 2. Some comments regarding special considerations for multiphase flows are included here. At impermeable walls, the usual 'no slip' boundary condition can be specified for the continuous phase. However, this condition will not be realistic for dispersed phase particles, which may slip along the wall if the particle size is bigger than the characteristic roughness scale of walls (Fig. 4.7). Sinclair and Jackson (1989) and Sommerfeld (1993) discussed the formulation of these boundary conditions in detail. For granular flows, Sinclair and Jackson (1989) formulated wall boundary conditions based on a microscopic model of particle collisions with the wall. They assumed that momentum flux transmitted to the boundary by collisions (product of the change of momentum per collision, collision frequency and the number of particles per unit area next to the wall) is equal to the tangential stress exerted by the particle adjacent to the wall. This leads to the following boundary condition:

where f is a specularity coefficient (equivalent to 1 minus the tangential restitution coefficient). For the granular temperature, a zero flux boundary condition is usually specified at the walls. Alternatively, it is possible to derive a boundary condition for granular temperature at the wall by equating the sum of the flux of granular temperature to the wall and the generation of granular temperature at the wall to the energy dissipation at the wall due to inelastic particle-wall collisions. Details may be found in Johnson and Jackson (1987). The other situation which may require special treatment is a boundary of multiphase dispersion through which dispersed phase particles are allowed to escape, but not the continuous phase (for example, the top surface of gas-liquid dispersion in a bubble column reactor). The standard 'outlet' boundary conditions need to be suitably modified to represent the observed flow processes. It is possible to simulate the actual behavior by specifying appropriate sink near the top surface (see Ranade, 1998 and Chapter 11).

So far, discussion has been restricted to laminar multiphase flow processes. However, in most industrial multiphase reactors, flow processes are turbulent. Modeling multiphase turbulent flow processes is a fast developing field. Some recent developments and general modeling issues are briefly discussed here. The presence and motion of dispersed phase particle may affect the macroscopic turbulence field of the continuous phase as well as the microscale characteristics of this turbulence field. Depending on particle size, density (particle response time) and volume fraction of the dispersed phase, dispersed phase particles may enhance or suppress the turbulence. Particles may interact with microscale fluctuations of continuous phase and may damp the turbulence (suspension and oscillations of particles may dissipate the energy). Larger particles may enhance turbulence by extracting energy from macroscale fluctuations and by vortex shedding. Elghobashi (1991) proposed a regime map based on particle response time and the Kolmogorov time scale to identify the influence (damping/enhancement) and degree of coupling between turbulence regimes in multiphase flows (Fig. 4.2). This map may be used to estimate the extent of coupling between dispersed phase particles and turbulence.

Most attempts at modeling complex, turbulent multiphase flows rely on the practices followed for single-phase flows, with some ad hoc modifications to account for the presence of dispersed phase particles. For multiphase flows, several new correlations appear in time-averaged (RANS type) governing equations. Different models have been proposed to specifically represent turbulence in dispersed gasliquid (reviewed by Ranade, 1995; Lahey, 1987; Lane et al., 1999) and dispersed gas-solid flows (reviewed by Sommerfeld, 1993; Bolio et al., 1995; Peirano, 1998). Some of these models are discussed in Part IV, while discussing the specific issues in modeling bubble column and fluidized bed reactors. The two-equation turbulence models (k-e) form the basis for most of these studies. Turbulence in a continuous phase is usually modeled using transport equations for turbulent kinetic energy and turbulent energy dissipation rates. These transport equations are written in a form similar to single-phase flows:

where 4>k can be turbulent kinetic energy or turbulent energy dissipation rate in phase k. The symbols /¿Tk and aTk have usual meanings of turbulent viscosity and turbulent Prandtl number for phase k. Spk is the corresponding source term for <p in phase k. Source terms for turbulent kinetic energy and dissipation can be written:

where Gk is generation in phase k and Gke is extra generation (or dissipation) of turbulence in phase k. Generation due to mean velocity gradients, Gk can be calculated using the single-phase flow equation (Eq. (3.23)). Extra generation or damping of turbulence due to the presence of dispersed phase particles is represented by appropriate formulation for Gke. Some formulations for Gke, suitable for gas-liquid and gas-solid flows, are discussed in Part IV when discussing different reactor types. In the absence of adequate information, in many cases, extra generation terms are usually set to zero. Various attempts to develop models to represent extra generation or damping of turbulence are reviewed by Lahey (1987) and more recently, by Peirano and Leckner (1998). The turbulent viscosity, /¿Tk , is calculated using the formula specified for single-phase flows (Eq. (3.20)). The values of parameters for the k-e model are usually kept the same as those for the single-phase model (listed in Table 3.1). In principle, values of these parameters will be functions of particle loading and ratio of particle relaxation time to eddy lifetime (Lahey, 1987; Squire and Eaton, 1994). Cao and Ahmadi (1995) discussed the variation of parameter (CM) appearing in the expression for turbulent viscosity with volume fraction of dispersed phase. More research is needed to predict the continuous phase turbulence in multiphase flows accurately. The situation becomes even more difficult when modeling turbulence in the dispersed phase. Turbulence in the dispersed phase may be physically understood as the particle velocity fluctuations caused by collisions between particles and interactions with the continuous phase. These modeling attempts are discussed by Balzer et al. (1995), Simonin (1995) and Enwald et al. (1996).

In most practical applications of dispersed multiphase flows, the suspension consists of non-spherical particles having different diameters. The range of particle diameters or particle size distribution may (non-reactive fluidization of particles) or

may not remain constant (bubble columns where gas bubbles may coalesce or break up and change the particle size distribution). It is necessary to adequately model the influence of particle size distribution on the fluid dynamics of multiphase flows. With the Lagrangian approach, the influence of particle size distribution can be readily included in the simulations. However, in a Eulerian-Eulerian approach, appropriate models and closures need to be developed. The three major approaches to accounting for the influence of particle size distribution (PSD) on fluid dynamics of multiphase flows are discussed below.

The first approach is an approximation, which relies on identifying an effective value of size of dispersed phase particle, which will behave in the same way as the population of dispersed phase particles with a specific PSD. Such an approach may give acceptable results if the PSD is narrow and smooth (uni-modal). There are many ways of defining a mean or an effective particle diameter for the suspension of particles. The mean volume-length diameter, the mean volume-surface diameter (Sauter mean diameter) and the mass mean diameter are the most commonly used mean diameters (Soo, 1990). According to Soo (1990), the mean volume diameter is suitable for the determination of volume fraction, whereas the mean surface diameter is relevant to physical mechanisms at the interface. Recently, Peirano and Leckner (2000) analyzed the fluid dynamics of gas-solid flows and concluded that for dilute suspensions (where collisional mechanisms can be neglected), if particle Reynolds numbers are smaller than 1, the mean volume-length diameter is the most representative. At higher Reynolds number (>1000), the mean volume-surface diameter is the most representative. For intermediate Reynolds number, the effective diameter smoothly varies from the mean volume-length to the mean volume-surface diameter. Such an effective diameter may be specified to simulate suspensions with narrow PSD.

When PSD is not smooth and wide (or is bi-modal), it is necessary to represent the dispersed phase by more than one fluid phase. The size distribution of dispersed phase particles is usually discretized into a few size groups. Each of these size groups is considered as an individual dispersed phase when simulating such multiphase flows. Several attempts have been made to develop a multiple solid phase granular flow model to simulate gas-solid flows with a wide size distribution of solid particles. Jenkins and Mancini (1987) extended the kinetic theory of granular flow to binary mixtures (assuming equal granular temperature). Gidaspaw et al. (1996) and Magner (1996) extended this to binary mixtures with unequal granular temperature. Recently, Mathiesen et al. (2000) developed a generalized gas-solid model with multiple solid phases. They also discuss several consistency issues in formulating such a generalized model. Such a generalized model requires appropriate formulations of particle-particle drag coefficients, apart from consistent formulations of various previously discussed terms such as radial distribution function, solids pressure, solids viscosity and so on. It must be noted that computational requirements increase significantly with an increase in the number of solid phases. Mathiesen et al. (2000) reported results of simulations with three solid phases, which show encouraging agreement with experimental data. This approach shows promise, and deserves further investigation to explore applications to industrially relevant multiphase flows.

A similar approach can be used to simulate fluid-fluid flows such as gas-liquid or liquid-liquid. However, in such flows the dispersed phase particles can coalesce or break up during the flow, and the particle size distribution evolves as the flow develops. Therefore, to define multiple phases with specific ranges of particle diameters to represent the dispersed phase, it is necessary to develop models of coalescence and break-up to simulate the changes in particle size distribution. Recently Carrica et al. (1999) reported the development of such a model to simulate bubbly two-phase flows. They derived conservation equations for different bubble phases by representing an equation of bubble size distribution function as a multi-group scheme. An equation for bubble number density corresponding to each group can then be derived after incorporating appropriate coalescence and break-up models (see for example, Prince and Blanch, 1990 or Luo and Svendsen, 1996 for coalescence and break-up models). Carrica et al. (1999) used 15 groups for simulating bubbly flow around a surface ship. Thus, instead of solving transport equations for two phases, they need to solve transport equations for 16 phases, which requires much larger computational resources. The required resources will be increased by a factor more than the ratio of the number of transport equations in the model (8), because of the increase in the required number of iterations to converge. When the bubble size distribution makes significant contributions, it is necessary to adopt such an approach. As an alternative to such intensive simulations, an approximate model for coalescence and break-up may be developed by considering a bubble number density equation. Coalescence and break-up processes locally affect bubble number density. A local effective bubble diameter can then be found from bubble number density and gas volume fraction. This approach avoids having to solve an excessive number of transport equations and may give adequate representation of the variation of effective bubble size within the domain. Lane et al. (1999) recently used such an approach to simulate gas-liquid flows in stirred vessels.

The recent progress in experimental techniques and applications of DNS and LES for turbulent multiphase flows may lead to new insights necessary to develop better computational models to simulate dispersed multiphase flows with wide particle size distribution in turbulent regimes. Until then, the simulations of such complex turbulent multiphase flow processes have to be accompanied by careful validation (to assess errors due to modeling) and error estimation (due to numerical issues) exercise. Applications of these models to simulate multiphase stirred reactors, bubble column reactors and fluidized bed reactors, are discussed in Part IV of this book.

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