## 43 Other Types Of Multiphase Flows

4.3.1. Flow Processes in Porous Media

Fixed bed reactors or packed bubble column reactors are examples of flow processes occurring in porous media, which are commonly encountered by reactor engineers. The problem of predicting fluid flow and the rates of mixing and other transport processes in such reactors is an important task for a reactor engineer. In general, two approaches are used to model the fluid dynamics of such reactors. The first one uses a lumped parameter approach and treats the irregular geometry of the packed region as an isotropic or non-isotropic porous media, characterized by a few lumped parameters. The second approach treats the geometrical intricacies of the packed region rigorously. Obviously, the second approach is computationally demanding and can be applied only to a small region of the reactor. It nevertheless can serve as a useful learning tool. These approaches are briefly discussed below.

In the lumped parameter approach, the packed region or the porous region is represented by introducing a suitable source or sink in the appropriate conservation equations. Most of the engineering simulations of flow through porous media (such as flow through fixed bed reactors or bundles of heat exchanger tubes) use this approach. The extra resistance offered by the porous medium to the flow can, for example, be represented by introducing a sink in the usual single-phase momentum conservation equations as:

where p is a permeability and C an inertial resistance factor characterizing the porous media. The values of these two empirical constants need to be specified either from experimental data or by using the more rigorous models mentioned earlier. To model a packed bed reactor, usually the Ergun (1952) equation is used to estimate the values of characteristic parameters. Use of the Ergun equation leads to the following expressions for p and C:

where Dp is particle diameter and w is porosity. It is possible to use different values of these characteristic constants for different directions to represent the anisotropy of porous media. Any other empirical pressure drop correlation may be used instead of the Ergun equation. To model the porous region of complex shaped particles, such as fibers of glass wool, more complex equations or a look-up table may also be used.

The additional sink is added to the usual conservation equations corrected for the volume fraction of the porous media. The governing equations look similar to those for Eulerian multiphase flow processes (Section 4.2.2) except that the volume fraction of the porous medium is not a variable. In the enthalpy equation, it is possible to include influence of porous media by considering an effective thermal conductivity, keff, of the form:

where w is a porosity. Subscripts f and s denote fluid and solid, respectively. It may also be necessary to include suitable modifications in the turbulence model to account for the different turbulence generation mechanism within the porous media. More often than not, the characteristic length scale of the porous region determines the characteristic length scale of the turbulence downstream of the porous region. The presence of porous media, therefore, decouples turbulence field upstream of porous region from the downstream. In such a situation, a length scale appropriate to the porous region under consideration may be used to estimate the desired turbulence quantities at the interface between the porous region and the downstream region.

Rigorous modeling of flow through porous media is very challenging, and recently, it has been explored as a learning tool. In this approach, microscopic flow processes are modeled in detail by considering a small periodic structure of the porous media. Geometrical details of solid regions and open regions are modeled rigorously. Therefore, although the governing equations become relatively simple (single-phase flow equations), the geometric modeling and grid generation become quite complicated. Computational demands also increase significantly. For example, Logtenberg et al. (1999) simulated flow and heat transfer in a fixed bed reactor by considering ten spherical particles in close contact. They had to use about 250 000 computational elements. Their model does not need any empirical parameters and could reproduce the flow and heat transfer behavior of a cluster of particles quite well. Such models will be very useful in developing a detailed understanding of microscopic flow structures in packed bed reactors and will complement lumped parameter models.

### 4.3.2. Separated Multiphase Flows

While discussing the flow through porous media in the above section, it was assumed that the open space is filled by a homogeneous continuous phase. It is possible to encounter situations in which more than one phase flows through packed beds. For example, in a typical absorption column/reactor, gas and liquid phase flow through the packed bed. For such a flow, there exists an interface between gas and liquid phases. Such a flow regime is called a separated flow regime. When gas and liquid phases are transported through conduits at high gas throughputs, an annular flow regime may exist in which liquid flows in the form of an annular film attached to the conduit walls and gas flows through the central core. This flow regime is also a separated flow regime. In a separated flow regime, several sub-regimes such as wavy flow may exist. To model such separated multiphase flows, it is necessary to use a volume of fluid (VOF) approach, since the interface between separated phases plays an important role. The basics of the VOF approach have already been discussed in Section 4.2.3. Application of VOF to simulate complex, separated multiphase flow have only recently started. Wehrli et al. (1997) and Casey et al. (1998) show some results of the application of VOF to simulate wavy flow over inclined solid surfaces. Considering the computational requirements of such simulations, the rigorous modeling of separated multiphase flows may be used as a learning tool and to developing semi-empirical lumped sub-models. Such lumped sub-models can then be used to simulate complex, multiphase flows through packed beds. The approach discussed for modeling flow through porous media may be extended to simulate multiphase flows through porous media. Details of modeling such packed bed or trickle bed reactors are briefly discussed in Chapter 13.

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