Interfacial Momentum Closure

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. 

However, to the author s knowledge, this model has not been applied for the prediction of multiphase reactors mostly due to the complexity of the suggested closure relations. On the other hand, this paper serves as a useful reference as the exact derivation of the k — e model equations are given and discussed. Parts of this modeling work have been adopted in many papers. 

Many authors have addressed the mathematical problem of averaging microscopic balance equations in order to derive macroscopic model formulations. However, the result is always a set of equations in which extra terms involving integrals over the microscopic domains remain. While various hypotheses may be made about interfacial closure laws expressing these extra terms as functions of the solution variables, it is not clear that such laws always exist, what form they should take and what approximations may be implied in their use. 

One can thus state that the constitutive equations for the interfacial terms are the weakest link in a multi-fluid model formulation because of considerable difEculties in terms of experimentation and modeling on a macroscopic level. The main difEculties in modeling arise from the existence of interfaces between phases and discontinuities associated with them. 

In practice one needs to provide appropriate formulations for the interfacial momentum closure laws using analytical expressions which are (to some extent) based on the general modeling principles presented in the introduction of this chapter, and that physically reproduce experimental results with reasonable fidelity. The formulations that will be given for various forces and effects will typically be applicable only for given ranges of particle-fluid conditions. 

---

The resulting decomposition of the interfacial momentum transfer term is equivalent to the conventional closure outlined in sect 3.4.3, and adopted by several investigations on gas solids flow [64, 65, 39, 108]. Nevertheless, as for the conventional formulation, several simplifying assumptions are invoked in... 

In practice, however, it is difficult to parameterize the interfacial area averaged velocity so in the momentum equation it s often set equal to the bulk velocity and in the momentum jump condition the mass transfer terms are simply neglected enabling a closure relation for the interfacial drag terms which are in agreement with Newton s 3. law. Hence,... 

---