Interfacial mass conservation equation

The derivation of the overall interfacial mass conservation equation is carried out in a fashion similar to the bulk mass conservation derivation in Section 3.1. The result will be of the same form as Eq. (3.1.4), except for the addition of a term corresponding to the jump in across the interface. With r the interface mass density (kgm ), the equation is... 

The intrinsic constitutive laws (equations of state) are those of each phase. The external constitutive laws are four transfer laws at the walls (friction and mass transfer for each phase) and three interfacial transfer laws (mass, momentum, energy). The set of six conservation equations in the complete model can be written in equivalent form ... 

For a given mass transfer problem, the above conservation equations must be complemented with the applicable initial and boundary conditions. The problem of finding the mathematical function that represents the behaviour of the system (defined by the conservation equations and the appropriate set of initial and boundary conditions), is known as a boundary value problem . The boundary conditions specifically depend on the nature of the physicochemical processes in which the considered component is involved. Various classes of boundary conditions, resulting from various types of interfacial processes, will appear in the remainder of this chapter and Chapters 4 and 10. Here, we will discuss some simple boundary conditions using examples of the diffusion of a certain species taken up by an organism ... 

In Eulerian-Eulerian (EE) simulations, an effective reaction source term of the form of Eq. (5.32) can be used in species conservation equations for all the participating species. The above comments related to models for local enhancement factors are applicable to the EE approach as well. It must be noted that interfacial area appearing in Eq. (5.32) will be a function of volume fraction of dispersed phase and effective particle diameter. It can be imagined that for turbulent flows, the time-averaged mass transfer source will have additional terms such as correlation of fluctuations in volume fraction of dispersed phase and fluctuations in concentration even in the absence... 

Our first major task is the description of the interfacial mass transfer process and, therefore, we shall examine further the equations for continuity of species i and the equation for conservation of total mass of mixture. 

The interfacial momentum exchange terms in the momentum conservation equations for each phase consist of drag and virtual mass force terms. The drag force for gas and liquid is modeled, respectively, as... 

The advantage of LS and D1 techniques over the VOF method is that their indicator functions are smooth, rather than discontinuous, and are easier to solve. Another advantage of the LS and D1 methods over the MAC method is that these two techniques do not suffer from the lack of divisibility that discrete particles exhibit. There are highly accurate numerical schemes that can be applied to the level set equation. The disadvantage of level set method, however, is that the level set needs to be reset periodically which is not strictly mass conservative. The DI method models interfacial forces as continuum forces by smoothing interface discontinuities and forces over thin but numerically resolvable layers. This... 

The stability analysis is very similar to that of Section 8 except that there are equations analogous to Equations 6.61 and 6.62 for both heat and mass transport in the liquid. Moreover, interfacial conservation equations both for energy and for the reactant A must be invoked. 

Clearly, then, we must know the concentration, temperature, and charge distributions at the interface in order to define the surface tension variation required to solve the hydrodynamic problem. However, these distributions are themselves coupled to the equations of conservation of mass, energy, and charge through the appropriate interfacial boundary conditions. The boundary conditions are obtained from the requirement that the forces at the interface must balance. This implies that the tangential shear stress must be continuous across the interface, and the net normal force component must balance the interfacial pressure difference due to surface tension. 

The general boundary condition for conservation of mass of some species in the interfacial region is derived in Chapter 6. For the present case of an insoluble surfactant, and in the absence of surface diffusion, we anticipate that the first two terms of Equation 5.32 should suffice with T replaced by A ... 

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