Solution of population balance equations using the FCMOM

The FCMOM (Finite size domain Complete set of trial functions Method Of Moments) is an efficient numerical technique to solve population balance equations. It was presented and validated for spatially homogeneous [1,3] and inhomogeneous conditions [2]. In this work, the general form of the FCMOM governing equations is presented which includes both the source terms and the terms accounting for spatially in-homogeneous conditions. Additionally, the role of the spatial diffusion terms in the FCMOM technique is investigated both as far as the reconstruction of particle size distribution partial derivative with respect to the internal variable and as far as the effect of the particle diffusivity models on the speed of propagation of the disturbances

Solution of PBE by MOM in Finite Size Domains

A new approach to solve PBE (Population Balance Equations), FCMOM (Finite size domain Complete set of trial functions Method Of Moments), is presented. The solution of the PBE is sought, instead of the [0, 1e] range, in the finite interval between the minimum and maximum particle size; their evolution is tracked imposing moving boundaries conditions. After reformulating the PBE in the standard interval [ 121, 1], the size distribution function is represented as a series expansion by a complete system of orthonormal functions. Moments evolution equations are developed from the PBE in the interval [ 121, 1]. The FCMOM is implemented through an efficient algorithm and provides the solution of the PBE both in terms of the moments and in terms of the size distribution function. The FCMOM was validated with applications to particle growth (constant, linear, diffusion-controlled), simultaneous particle growth and nucleation, particle dissolution, particle aggregation (constant, sum, product, Brownian kernels) and simultaneous particle aggregation and growth

Estudo sobre o nivel de participação, num programa de atividade fisica e saude e suas relações com as doenças cronicas não transmissiveis e a qualidade de vida : um estudo de caso

Orientador : Antonia Dalla Pria BankoffTese (doutorado) - Universidade Estadual de Campinas, Faculdade de Educação FisicaResumo: A saúde e a doença acompanham o ser humano ao longo da história da humanidade. Ao buscar saúde para não: ser, estar ou ficar doente busca também a qualidade de vida, mas com toda sua complexidade e capacidade de criar, cria também o sedentário e acaba gerando novas doenças que lhe tiram a saúde. Hoje em dia se busca a qualidade de vida para combater o mal criado pelo próprio ser humano. O objetivo deste trabalho foi o de caracterizar os funcionários da Diretoria Geral Administrativa da Universidade Estadual de Campinas com a intenção de implantar um programa de atividade física orientada visando a qualidade de vida. Fizemos também uma investigação sobre as preocupações da educação física em relação ao ser humano, com a saúde, a doença e os diversos programas de saúde que buscaram e buscam a qualidade de vida. Os procedimentos metodológicos utilizados foram à pesquisa da observação direta extensiva quantitativa e a qualitativa, através da pesquisa ação, por apresentar os requisitos necessários a tais procedimentos utilizamos como instrumento de pesquisa dois protocolos com questões abertas e fechadas: uma para a caracterização dos participantes e outro para identificar os motivos da não aderência ao programa oferecido, mas como o ser humano é complexo e mesmo sabendo de suas vantagens não adere ao programa. Os resultados foram apresentados através de gráficos seguidos das análises das respostasAbstract: Sickness and health walk side by side with the human race alI through out its history. Along the human search for health in order not to be or get sick he also seeks life quality, but due to its complexity and capacity of creation, he also creates the sedentary and ends up generating new deseases that take his health away. In our days we search quality of life in order to fight the bad created by the human being. The objective of this work was to characterize the employees of the Diretoria Geral Administrativa da Universidade Estadual de Campinas (General Administrative Directory of the Campinas University) with the intention of implanting a physical activity program oriented to life quality. However, since the human being, as mentioned before, is very complex, even knowing about the benefits of the program the employess did not adehere to it We also investigated existing programs that seek lHe quality as well as Physical Education worries about the human being health and disease. The methodological procedures used were the research of the direct quantitative and qualitative extensive obsevation, through action research. We used as research instruments two protocols with open and closedended questions: one for the participants charaderization and another to identify the reasons of the non-adeherence to the proposed programo The results were presented through graphics and analysis of the answersDoutoradoDoutor em Educação Físic

Solution of bivariate population balance equations using the FCMOM

The FCMOM (Finite size domain Complete set of trial functions Method Of Moments) is an efficient and accurate numerical technique to solve PBE (population balance equations) and was validated for monovariate PBE [Strumendo, M.; Arastoopour, H. Solution of PBE by MOM in Finite Size Domains. Chem. Eng. Sci. 2008, 63 (10), 2624]. In the present paper, the FCMOM is extended and used to solve bivariate PBE. In the FCMOM, the method of moments is formulated in a finite domain of the internal coordinates and the particle distribution function is represented as a truncated series expansion by a complete system of orthonormal functions. In the extension to bivariate PBE, the capabilities of the FCMOM are maintained, particularly as far as the algorithm efficiency and the accuracy in the bivariate particle distribution function reconstruction. The FCMOM was validated with the following bivariate applications: particle growth, particle dissolution, particle aggregation, and simultaneous aggregation and growth

Solution of population balance equations by the FCMOM for in-homogeneous systems

The FCMOM (finite size domain complete set of trial functions method of moments) is an efficient and accurate numerical technique to solve monovariate and bivariate population balance equations. It was previously formulated for homogeneous systems. In this paper, the FCMOM approach is extended to solve monovariate population balance equations for inhomogeneous (spatially not uniform) systems. In the FCMOM, the method of moments is formulated in a finite domain of the internal coordinates and the particle size distribution function is represented as a truncated series expansion by a complete system of orthonormal functions. The FCMOM is extended to inhomogeneous systems assuming that the particle-phase convective velocity is independent of the internal variables (particle size). The method is illustrated by applications to particle diffusion and to particle convection. In the case of particle convection, a gas-solid dilute flow in a pipe was simulated and the FCMOM equations were coupled with the governing equations (mass and momentum balances) of the gas phase

Method of moments for the dilute granular flow of inelastic spheres

Some peculiar features of granular materials (smooth, identical spheres) in rapid flow are the normal pressure differences and the related anisotropy of the velocity distribution function f (1). Kinetic theories have been proposed that account for the anisotropy, mostly based on a generalization of the Chapman-Enskog expansion [N. Sela and I. Goldhirsch, J. Fluid Mech. 361, 41 (1998)]. In the present paper, we approach the problem differently by means of the method of moments; previously, similar theories have been constructed for the nearly elastic behavior of granular matter but were not able to predict the normal pressures differences. To overcome these restrictions, we use as an approximation of the f (1) a truncated series expansion in Hermite polynomials around the Maxwellian distribution function. We used the approximated f (1) to evaluate the collisional source term and calculated all the resulting integrals; also, the difference in the mean velocity of the two colliding particles has been taken into account. To simulate the granular flows, all the second-order moment balances are considered together with the mass and momentum balances. In balance equations of the Nth-order moments, the (N+1)th-order moments (and their derivatives) appear: we therefore introduced closure equations to express them as functions of lower-order moments by a generalization of the \u2018\u2018elementary kinetic theory,\u2019\u2019 instead of the classical procedure of neglecting the (N+1)th-order moments and their derivatives. We applied the model to the translational flow on an inclined chute obtaining the profiles of the solid volumetric fraction, the mean velocity, and all the second-order moments. The theoretical results have been compared with experimental data [E. Azanza, F. Chevoir, and P. Moucheront, J. Fluid Mech. 400, 199 (1999); T. G. Drake, J. Fluid Mech. 225, 121 (1991)] and all the features of the flow are reflected by the model: the decreasing exponential profile of the solid volumetric fraction, the parabolic shape of the mean velocity, the constancy of the granular temperature and of its components. Besides, the model predicts the normal pressures differences, typical of the granular materials