Christophe Morel

Surface Equations for Two-Phase Flows

This chapter is devoted to the presentation of the fundamental equations governing the interfacial surfaces. We begin by recalling the definitions of the different kinds of interfacial areas: the global one and the local one and the existing link between them. We pursue by a presentation of the different forms of the Leibniz rule (or Reynolds transport theorem) for a surface. The interfacial area balance equation can be understood as a particular case of this Leibniz rule, except for the discontinuous phenomena like coalescence and breakup which must be added for completeness. The average equations for the void fraction and for the interfacial area are then derived and their closure issue is examined. For strongly non-spherical interfaces (e.g. for strongly deformed bubbles or droplets), the area tensors are introduced, which are a new tool to deal with the tensorial aspect of non-spherical interfaces. The interfacial area balance is then completed by an additional transport equation for the second order area tensor or for its deviator, which is named the interface anisotropy tensor.

Closures for the Bubble Size Distribution and Interfacial Area Concentration

This chapter is devoted to the presentation of the closure laws for the interfacial area transport equation and other multi-size bubble models. Considering first the single size case, the different forms of the interfacial area transport equation are recalled and their closure laws are reviewed. These closure laws concern essentially the interfacial area variations due to the coalescence, breakup and phase change phenomena. The gas expansion as well as the nucleation and collapse are also considered. In the second part of the chapter, we present some possible closures for the more difficult case of multi-size bubbly flows. Two approaches are followed: the moment’s method with a presumed size NDF and a class method using a discretization of the NDF. In the moment’s method, two different mathematical expressions are used for the NDF: a log-normal law and a quadratic law.

Macroscopic Formulation of Two-Phase Flows: The Two-Fluid Model

This chapter is devoted to the averaging of the equations derived in Chap. 2, thus obtaining the equations at the macroscopic level . Two versions of the two-fluid model are derived. The first one is very general and has been derived by numerous authors a long time ago; we call it the classical two-fluid model. The second version of the two-fluid model is more recent and is devoted to the analysis of disperse flows only. In this model, the balance equations written for the disperse phase reflect the equations written for a single particle, and are slightly different from the equations written for the continuous phase. This dissymmetry between the equations written for the two phases reflects the real dissymmetry between the phase’s geometries (one continuous and one disperse). This second version is called the hybrid two-fluid model. We give a comparative analysis of the closure problem posed by the two models and analyze these closure issues with the help of the second law of thermodynamics.

Interfacial Heat and Mass Transfers

This chapter is a short introduction to the modeling of the interfacial heat and mass transfers. First of all, we derive an approximate relation between the interfacial transfer of mass and the two interfacial heat transfers between the two phases and the interface. This approximate relation is used in numerous studies on vapor-liquid flows and we show what approximations must be done to obtain it. After then, the liquid-to-interface heat transfer is modelled in the case of bubbly flows. The two cases of conductive and convective heat transfers are clearly separated. The chapter ends with a brief presentation of the modeling of the vapor-to-interface heat transfer in the case of droplet flows.

Interfacial Forces and Momentum Exchange Closure

This chapter gives an introduction to the modeling of the different forces acting on a disperse particle, and exerted on it by the continuous fluid. Different academic situations are first examined: spherical particles, very small or very large Reynolds numbers… The forces acting on a particle are decomposed into two contributions. The first contribution comes from the unperturbed fluid (by the particle presence) and is constituted of the Archimedes and Tchen forces. The second contribution comes from the perturbations and is classically decomposed into the sum of the drag, added mass, lift, and history forces. The proximity of a wall gives an additional lubrication force which is called the wall force. The effects of the particles shape and concentration are also examined in Sect. 8.6. In the last Sect. 8.7, the mean momentum interfacial transfer term is derived from the knowledge of the different forces and a proper averaging procedure.

Example of Application: Bubbly Flow in a Vertical Pipe

This chapter gives an example of application which concerns a bubbly flow in a vertical pipe. First of all we summarize the balance equations and a set of closure relations for them. The choice of the closure relations does not really matter and other choices could be done. The flow is assumed to be upwardly directed in a vertical pipe having a circular cross section. Due to this particular geometry, we choose to project the equations in a cylindrical coordinates system and assume the flow to be axisymmetric. The cylindrical coordinates are a special kind of more general curvilinear coordinates. After summarizing some elements of the theory of curvilinear orthogonal coordinate systems, we come back to the particular case of cylindrical coordinates and project our equations in this coordinates system. Then, we discretize all the equations by using a method proposed par Patankar (1980) and indicate how to solve them.

Turbulence Equations for a Disperse Phase

This chapter deals with turbulence modeling for a disperse phase embedded in a continuous one. By disperse phase turbulence, we do not mean the turbulence existing in the interior of fluid particles (if any). Instead, we mean the fluctuating motion of the particles themselves, each particle having its own velocity that is different from the mean velocity of the swarm. In order to do this kind of turbulence description, a two-particle Number Density Function (NDF) is introduced (one continuous fluid particle and one discrete particle). Real particles are replaced by “stochastic particles” which are assumed to give the same statistics than the real particles. A Langevin modeling approach is also introduced to model the fluid velocity seen by the particles. All the turbulence equations for the two phases are rewritten according to this formalism, and the compatibility relations between the equations of this chapter and the equations derived in the previous chapters are indicated. The present chapter ends with the introduction of the inter-particles collisions in some of the equations derived, and a summary of the closure issues.

Microscopic Formulation of Two-Phase Flows

The equations governing a two-phase flow at the microscopic level are presented. By microscopic level, we mean that no averaging operation is done. The equations at the microscopic level are local and instantaneous equations. We begin by presenting the topological equation, followed by the mass, momentum and energy balance equations. The entropy equation is also presented is order to examine the consequences of the second law of thermodynamics. We end this chapter by deriving the equations which will be useful for the derivation of the two-fluid models in Chap. 3. The balance equations governing a whole fluid particle (i.e. a bubble or a droplet) are also given for completeness.

Population Balances and Moments Transport Equations for Disperse Two-Phase Flows

In this chapter, a link is derived between the interfacial area transport equation derived in Chap. 4 and the population balance equation for disperse flows. The particles birth and death phenomena are formalized. These phenomena are mechanical ones (coalescence and breakup) or are due to phase change (nucleation and collapse). Transport equations are derived for the moments of the particle size distribution function. These moments equations being unclosed, we briefly review two quadrature methods of moments to close and solve the set of equations. A completely different method based on the discretization of the bubble size distribution function is also presented in the context of isothermal bubbly flows.

Turbulence Equations for a Continuous Phase

A complete derivation of the turbulence equations for a continuous phase is given in the present chapter. We first recall the derivation of the turbulence equations for a single phase flow. Then these equations are extended to the continuous phase of a two-phase flow. In each case, the derived equations are the equations for the mean motion (mass and momentum), for the Reynolds stress tensor, the turbulent kinetic energy, the turbulence dissipation rate and the turbulence equations governing a passive scalar like the temperature or a species concentration. We end this chapter by summarizing the closure issues in the single phase case as well as in the two-phase one.