Locally homogeneous flow models

Of major interest concerning these problems are influences of turbulence in spray combustion [5]. The turbulent flows that are present in the vast majority of applications cause a number of types of complexities that we are ill-equipped to handle for two-phase systems (as we saw in Section 10.2.1). For nonpremixed combustion in two-phase systems that can reasonably be treated as a single fluid through the introduction of approximations of full dynamic (no-slip), chemical and interphase equilibria, termed a locally homogeneous flow model by Faeth [5], the methods of Section 10.2 can be introduced reasonably successfully [5], but for most sprays these approximations are poor. Because of the absence of suitable theoretical methods that are well founded, we shall not discuss the effects of turbulence in spray combustion here. Instead, attention will be restricted to formulations of conservation equations and to laminar examples. If desired, the conservation equations to be developed can be considered to describe the underlying dynamics on which turbulence theories may be erected—a highly ambitious task. 

Most current multidimensional spray simulations have adopted the thin or very thin spray assumptions,[55°1 i.e., the volume occupied by the dispersed phase is assumed to be small. This can be justified if a simulation starts some distance downstream of the nozzle exit, where the gas volume fraction is large enough, or if the computational cells are relatively large. Accordingly, two major classes of models have been used in spray modeling locally homogeneous flow (LHF) models and two-phase-flow or separated-flow (SF) models. 

Figure 5.16. The four-environment model divides a homogeneous flow into four environments, each with its own local concentration vector. The joint concentration PDF is thus represented by four delta functions. The area under each delta function, as well as its location in concentration space, depends on the rates of exchange between environments. Since the same model can be employed for the mixture fraction, the exchange rates can be partially determined by forcing the four-environment model to predict the correct decay rate for the mixture-fraction variance. The extension to inhomogeneous flows is discussed in Section 5.10.

To quantify this treatment of migration as influenced by kinetics, a model has been developed in which instantaneous or local equilibrium is not assumed. The model is called the Argonne Dispersion Code (ARDISC) ( ). In the model, adsorption and desorption are treated independently and the rates for adsorption and desorption are taken into account. The model treats one dimensional flow and assumes a constant velocity of solution through a uniform homogeneous media. 

One of the earliest models for turbulence modulation in homogeneous dilute particleladen flows is Hinze s model (1972), in which the assumption of vortex trapping of particles is employed. On the basis of this model, the particle turbulent kinetic energy, kp, is determined by the local gas turbulent kinetic energy k as... 

As our first example, we consider the case of a first-order homogeneous reaction A -> B in a laminar flow tubular reactor for which the global equation is linear in c (i.e. r( c)) — (c)) and is therefore completely closed. To obtain the range of convergence of the two-mode model, we need to consider only the local equation. In this specific case, the reduced model equations to all orders of p are then given by... 

A more rigorous pseudo-homogeneous two-dimensional axi-symmetric model can be obtained reducing the governing averaged equations, that can be derived using any of the local averaging procedures described in sect 3.4, for the particular axi-symmetric tube flow problem. 

It is also assumed that ionic species or their ionic pairs cannot diffuse in the membrane phase without intervention by the carrier. The membrane is formally divided into n homogeneous compartments which contain the carrier in the form C2A, CqB, and CH the local concentrations of which are equal to [C2A]j, [C2B]., and [CH], with k= 1, 2,..In the case of bulk agitated or flowing liquid membranes, the model should be modified by adding a large central compartment (layer). 

In the more sophisticated treatments, spatial inhomogeneities are modeled by connected zones or regions of Space that are assumed to be homogeneous. The resulting population-balance equation (PBE) does not explicitly account for local variations in die flow field. 

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