Microscopic transport equation

The microscopic transport equations for the reaction-progress variables can be found from the chemical species transport equations by generalizing the procedure used above for the acid-base reactions (Fox, 2003). If we assume that Fa Fb Fc, then the transport equations are given by... 

As a first example of a CFD model for fine-particle production, we will consider a turbulent reacting flow that can be described by a species concentration vector c. The microscopic transport equation for the concentrations is assumed to have the standard form as follows ... 

This system of 2 M nonlinear equations is ill-conditioned for large M, but can be efficiently solved using the product-difference (PD) algorithm introduced by McGraw (1997). Thus, given the set of 2 M moments on the left-hand side of Eq. (107), the PD algorithm returns wm and lm for m — 1., M. The closed microscopic transport equation for the moments can then be written for k — 0,..., 2 M— 1 as... 

The simplest aggregation and breakage models can be formulated in terms of the NDF (o), which uses volume as the independent variable.6 The microscopic transport equation for the NDF has the form (Wang et al., 2005a,b)... 

To overcome the difficulty of inverting the moment equations, Marchisio and Fox (2005) introduced the direct quadrature method of moments (DQMOM). With this approach, transport equations are derived for the weights and abscissas directly, thereby avoiding the need to invert the moment equations during the course of the CFD simulation. As shown in Marchisio and Fox (2005), the NDF for one variable with moment equations given by Eq. (121) yields two microscopic transport equations of the form... 

The extension of DQMOM to bivariate systems is straightforward and, for the surface, volume NDF, simply adds another microscopic transport equation as follows ... 

The microscopic transport equations are often considered too complex for practical applications considering reactor optimization, scale-up and design. In engineering practice integral averages of the equations over one, two or three spatial directions are used reducing the model to that of an ideal reactor model type. 

It is seen that the governing system equation (3.51) is transformed into a generic Eulerian control volume formulation (3.81) consisting of a volume integral determining the microscopic transport equations for the bulk phases and a surface integral determining the jump balance at the interface. 

The Microscopic Transport Equations for a Finite Number of Dispersed Phases - the Multi-Fluid Model... 

The thermodynamics of irreversible processes begins with three basic microscopic transport equations for overall mass (i.e., the equation of continuity), species mass, and linear momentum, and develops a microscopic equation of change for specific entropy. The most important aspects of this development are the terms that represent the rate of generation of entropy and the linear transport laws that result from the fact that entropy generation conforms to a positive-definite quadratic form. The multicomponent mixture contains N components that participate in R independent chemical reactions. Without invoking any approximations, the three basic transport equations are summarized below. 

We shall be interested in a broad, macroscopic, view of reactors—one where elementary mass balances are applied over entire process units. That is, we are not interested in modeling the system in detail, and attainable region (AR) theory does not demand the use of microscopic transport equations for instance. 

General microscopic transport equations can be found in standard text books on transport phenomena (see the bibliography). For example, the transient 3D binary mass transport equation in rectangular coordinates is obtained by using a differential control volume in three dimensions, as depicted in Figure 3.3. 

---