Finite Size Domain Complete set of Trial Functions Method of Moments (FCMOM) Approach
In all of the solution methods based on the method of moments (MOM), the key is writing the transport equation (PBE) in terms of the lower-order moments of the number density function, f, in a closed form (Marchisio et al. 2003a). The ith moment of the number density function is defined as
Using FCMOM, the solution of PBE is sought in the finite interval between the minimum and maximum values of the particle property (e.g., size), instead of in the [0,i] range (Strumendo and Arastoopour 2008). In general form, the evolution of the number density function is tracked by imposing moving boundary conditions. After reformulating PBE in the standard interval [—1, +1] by a coordinate transformation as Strumendo and Arastoopour (2008),
where the dimensionless (divided by an appropriate scale factor) size distribution function f (|; x, f) is represented as a series expansion by a complete system of orthonormal functions (e.g., Legendre polynomials). Writing the distribution function in terms of a series expansion of Legendre polynomials gives
where the coefficient cn e can be expressed in terms of the moments
and where the terms with negative moments order (2v — n < 0) are omitted (Strumendo and Arastoopour 2008).
The orthonormal functions, фп (<*), associated with the Legendre polynomials
Therefore, a set of transport equations for the moments of the distribution function, f, could be derived from the general PBE (4.1) in the interval of [—1, +1], as presented in Eq. (4.9):
On the right-hand side of the moments evolution, Eq. (4.9), the first five terms are due to the coordinate transformation, IG is due to the Integration of the Growth term, and S is the source term due to the aggregation and breakage. These terms are:
In the above terms, the subscripts of—1 and +1 refer to the value of that term at the minimum and maximum limits of the range [—1, +1], respectively.
In the derivation of Eq. (4.9), it has been assumed that the particulate phase is
incompressible, i.e., = 0, where vp is the particulate phase velocity. Moreover,
it is assumed that the particles are convected with an average phase velocity vp(x,t), which means the particle velocity is independent of the internal property (e.g., size). In this case, the need for a spatial diffusion term would arise (Mazzei 2013). Mazzei (2011, 2013) has investigated the importance of diffusion when dealing with segregation dynamics of polydisperse systems and has shown that either a size- dependent velocity or a spatial diffusion term is necessary to model the segregation in polydisperse systems. He proposed a method to replace the average phase velocity vp(x,t) with a size-conditioned velocity field. In this case, he showed that, because the advective term features a size-dependent velocity field, the equation presents no diffusive flux in physical space. Having that in mind and for the sake of brevity, the diffusive term was omitted from the governing equation, that reduces Eqs. (4.9-4.17):
S is the source term and accounts for the introduction of new particles into the system, which we assume is due only to the aggregation as defined by Eq. (4.2). To handle this term using FCMOM, it is necessary to define a finite version of the Smoluchowski equation as proposed by Strumendo and Arastoopour (2008):
In the finite version of the Smoluchowski equation, a minimum and maximum size, ?min and ?max, are set, and aggregations leading to particles larger than ?max are neglected by using ?max — ? as the upper limit of integration in the second term on the right-hand side of Eq. (4.13), by introducing the Heaviside step function H. By choosing values of ?max large enough and setting ?min = 0, the solution of the finite version of the Smoluchowski equation converges to the solution of its classical version (Eq. 4.2).
Therefore, the dimensionless form of the source term, S, becomes Emni and Fmni are coefficients that can be pre-calculated as a function of ?min and
In general, moments evolution equations must be coupled with the moving boundary conditions providing the governing equations for ?min(f, x) and ?max(f,x). The moments evolution equations and the moving boundary conditions form a set of partial differential equations when the variables are the moments of the distribution function ^;(f, x), and two moving boundaries, ?min(f,x) and ?max(f, x). The initial conditions for the moments are computed from the initial property (e.g., size) distribution function. However, in the case of pure aggregation in which ?min and ?max are set initially to constant values, the problem is no longer a moving boundary problem, and the terms MB and MBConv will become zero. In this case, the final form of the moment transport equation is
A detailed explanation of the FCMOM method and derivation of the governing equation can be found in two papers by Strumendo and Arastoopour (2008, 2010).