Disperse multiphase flow dilute

Frequently encountered in nature and process industries, multiphase flows may comprise various states of matter, e.g., gas and solid in fluidization gas and liquid in bubble column and gas, liquid, and solid in airlift slurry bed (Mudde, 2005). In this article, the term phase in multiphase flow is related to the aggregative state of flow, which is normally far from equilibrium states. And it is different from the phase for a thermodynamic equilibrium system, where the phase is used to refer to a set of equilibrium states that can be demarcated in terms of state variables by a phase boundary on a phase diagram. As a result, it is possible to have a gas—soHd flow mixture with more than two phases, which can be classified by size, density of particles, or by the states of dispersion, e.g., poly disperse multiphase flow pCue and Fox, 2014) and dilute—dense, gas—soHd multiphase flow (Hong et al, 2012). 

For multiphase systems a rough distinction can be made between systems with separated flows and those with dispersed flows. This classification is not only important from a physical point of view but also from a computational perspective since for each class different computational approaches are required. For multiphase systems involving multiphase flow both Eulerian, mixed Eulerian-Lagrangian, and two-material free surface methods can be used. An excellent review on models and numerical methods for multiphase flow has been presented by Stewart and Wendroff (1984). A similar review with emphasis on dilute gas-particle flows has been presented by Crowe (1982). 

In problems in which the dispersed phase momentum equations can be approximated and reduced to an algebraic relation the mixture model is simpler to solve than the corresponding multi-fluid model, however this model reduction requires several approximate constitutive assumptions so important characteristics of the flow can be lost. Nevertheless the simplicity of this form of the mixture model makes it very useful in many engineering applications. This approximate mixture model formulation is generally expected to provide reasonable predictions for dilute and uniform multiphase flows which are not influenced by any wall effects. In these cases the dispersed phase elements do not significantly affect the momentum and density of the mixture. Such a situation may occur when the dispersed phase elements are very small. There are several concepts available for the purpose of relating the dispersed phase velocity to the mixture velocity, and thereby reducing the dispersed... 

Zhang DZ, Prosperetti A (1997) Momentum and Energy Equations for Disperse Two-Phase Flows and Their Closure for Dilute Suspensions. Int J Multiphase Flow 23(3) 425-453... 

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Sachdev, J. S., Groth, C. P. T. Gottueb, J. J. 2007 Numerical solution scheme for inert, disperse, and dilute gas-particle flows. International Journal of Multiphase Flow 33, 282-299. 

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The multi-fluid models discussed in Chap. 3 have been employed extensively describing the behavior of cold flow and reactive flow systems in multiphase reactors. However, for gas-solid fluidized bed systems, and in particular for dense systems, it has been found that the dispersed phase behavior cannot be captured without an extended description of the pressure tensor closure. The first extensions of the dilute continuum mechanical two-fluid model concept included a very rough approximation of the solid phase collisional pressure in terms of a semi-empirical correlation for the modulus of elasticity and a constant representative particle fluid viscosity. This type... 

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