Quadrature aggregation

Marchisio, D. L., R. D. Vigil, and R. O. Fox (2003). Quadrature method of moments for aggregation-breakage processes. Journal of Colloid and Interface Science 258, 322-334. 

Marchisio DL, Vigil RD Fox RO (2003) Implementation of the quadrature method of moments in CFD codes for aggregation-breakage problems. Chem Eng Sci 58(15) 3337-3351... 

In the quadrature method of moments (QMOM) developed by McGraw [131], for the description of sulfuric acid-water aerosol dynamics (growth), a certain type of quadrature function approximations are introduced to approximate the evolution of the integrals determining the moments. Marchisio et al [122, 123] extended the QMOM for the application to aggregation-breakage processes. For the solution of crystallization and precipitation kernels the size distribution function is expressed using an expansion in delta functions [122, 123] ... 

Because the accuracy of the quadrature approximation strongly depends on the application, let us discuss some of these issues in detail for specific examples. In the case of standard nucleation, (positive) growth, aggregation, and breakage, application of the QBMM to the source term in Eq. (7.96) yields... 

Silva, L. E. L. R., Rodrigues, R. C., Mitre, J. E. Lage, R L. C. 2010 Comparison of the accuracy and performance of quadrature-based methods for population balance problems with simultaneous breakage and aggregation. Computer and Chemical Engineering 34, 286-297. 

Monteagudo et al characterized the asphaltenes as a continuous ensemble for which the distribution function was taken from the fractal aggregation theory. The asphaltene family was discretized in pseudo-components by the Gauss-Laguerre quadrature. Only the asphaltene polydispersity was taken into account. All other components were represented by as solvent whose properties (molar volume and solubility parameter) were calculated form a cubic equations of state. Aggregation of asphaltenes was considered to be a reversible process. And it was assumed the phase equilibrium was between a liquid phase and a pseudo-liquid phase containing only asphaltenes. 

In the quadrature method of moment (QMOM) a few moments of the number distribution function/ are tracked in time directly, just as for the standard MOM, but in this approach the requirement of exact closure is replaced by an approximate closure condition that allows the method to be applied under a much broader range of conditions. This method was first proposed by McGraw [151] for modeling aerosol dynamics and has later been extended to aggregation and breakage processes in crystallization by Marchisio et al. [141, 142]. 

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