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Eulerian modelingand computational ﬂuid dynamics simulation of mono andpolydisperse ﬂuidized suspensions
Luca Mazzei
Department of Chemical Engineering, University College London, Torrington Place, London WC1E 7JE, UK
August 29
th
, 2008
Luca Mazzei
Abstract
1
1 Abstract
Fluidization, the operation consisting in passing a ﬂuid upwards through a bed of particles so as toconfer the latter a ﬂuid-like behavior, is a winning technology used in several industrial processes.Designing ﬂuidized bed, however, has always been extremely complex, since the behavior of thesesystems is difﬁcult to predict. Computational ﬂuid dynamics (CFD) permits to investigate ﬂuidizedbeds without having to resort to pilot plants and scaling-up empirical relations; it is nonethelesscritical that the ﬂuid dynamic models be sufﬁciently accurate. There are essentially two approachesto model multiphase systems: the
Eulerian-Lagrangian
, which tracks the motion of each particle andsolves the dynamics of the ﬂuid at a length scale much smaller than the particle diameter (microscopiclength scale), and the
Eulerian-Eulerian
models, based on averaged equations of motion which treatthe ﬂuid and solid phases as interpenetrating continua. Being computationally less demanding, thissecond approach is often preferred, especially by the industry. Its major drawback, however, is thatthe equations of motion are not mathematically closed and appropriate constitutive relations must bedeveloped on semitheoretical grounds.My PhD was primarily concerned with the Eulerian-Eulerian mathematical modeling of ﬂuidizedsuspensions. I started it by deriving rigorously the space-averaged equations of motion for particulatesystems of
n
particle classes (Owoyemi et al., 2007; Mazzei, 2008); this provides considerable insight
into the mathematical srcin of the various contributions featuring in said equations and is necessaryto attain a well-posed multiphase model.I then developed constitutive relations required to close the ﬂuid-particle interaction force, withparticular emphasis on two contributions: the buoyancy force and the drag force.First, I examined the buoyancy force, comparing the classical deﬁnition, which regards it as equalto the weight of ﬂuid displaced by the particles, to alternative deﬁnitions used in multiphase ﬂows. Ialso explained why the classical deﬁnition should be preferred (Mazzei & Lettieri, 2007).
Then, I analyzed critically several closure relationships available in the literature used to modelthe drag force in monodisperse systems (Mazzei & Lettieri, 2007, 2008). To put the validity of these
closures to the test, I studied the expansion proﬁles of homogeneous ﬂuidized beds both analyticallyand computationally, and I compared the results with experimental data. The analysis showed thatno relationship is entirely consistent with the empirical correlation of Richardson & Zaki (1954); in
fact, in some ﬂuid dynamic regimes and for some values of the voidage, the predictions were quiteat variance. Since this correlation is particularly reliable and accurate, I developed and tested a newequation of closure entirely based on it (Mazzei & Lettieri, 2007); consistency was thus obtained
over the whole range of ﬂuid dynamic regimes and for any value of the ﬂuid volume fraction. Thisproperty is essential when simulating the motion of ﬂuidized suspensions as it ensures amore accurateprediction of the expansion proﬁles of homogeneous systems, a feature that indirectly reﬂects a betterassessment of the drag force magnitude.Successively, Iinvestigated thestability ofparticulate ﬂuidized systems; thiswasdone analyticallyby means of linear stability analysis (Mazzei et al., 2006). I observed that if the only ﬂuid-particle
interaction forces at play are the buoyancy and the drag, formal stability is not possible – the sameconclusion holds if additional terms, such as the virtual mass force or the lift force, are considered.This seems to contradict experimental evidence. To overcome this inconsistency, following Foscolo& Gibilaro (1987), I introduced a novel force, named
elastic force
. Its origin is based on purelyphysical considerations and is related to void fraction gradients arising when the system homogeneityis lost. The closure that I advanced, as opposed to that developed by the aforementioned authors, is
Luca Mazzei
References
2not limited to conditions approaching equilibrium and holds in a much more general framework, aproperty which renders it suitable for the CFD study of multiphase ﬂows.By putting together these results, I proposed a new Eulerian-Eulerian ﬂuid dynamic model forﬂuidized suspensions that features novel formulations for all the main contributors to the ﬂuid-particleinteraction force (Mazzei & Lettieri, 2008).
The model just described caters for monodisperse systems. As previously pointed out, however,an important property of multiphase particulate systems is the particle size distribution of the granularmaterial making up the suspension. To improve the modeling, we must therefore treat the system aspolydispersed, considering the presence of several discontinuous phases interacting with one anotherand with the ﬂuid. To this end, I adopted a more powerful modeling approach to derive suitableequations of change for the disperse phases (Mazzei, 2008). To describe their evolution, I resorted
to the generalized population balance equation (GPBE): a continuity statement written in terms of particle velocity and additional coordinates, such as the particle diameter.Finding the solution of GPBEs usually requires a detailed description of the ﬂuid dynamics andof the interactions between mixing and chemical reactions (if present); for this reason, CFD hasbecome a standard tool for modeling this type of systems. Solving GPBEs within CFD codes is avery interesting and difﬁcult subject, and in recent years many approaches have been considered. Themain difﬁculty resides in the dimensionality of the equation: whereas classical transport equationsare three-dimensional, the GPBE is usually written in a higher-dimensional space and consequently isincompatible with customary (
i
.
e
., three-dimensional) computational schemes. To solve the equation,I used the
method of moments
(MOM). This requires the use of a limited number of moments of theGPBE to derive three-dimensional transport equations. The limited set of equations, which replacesthe single multidimensional population balance equation, keeps the problem tractable when appliedto complicated applications in multiphase ﬂows. The main obstacle to the method is that the momenttransport equations are mathematically unclosed.To overcome the problem, I employed two very efﬁcient methods, the
direct quadrature method of moments
(DQMOM) and the
quadrature method of moments
(QMOM). Both approximate thevolume density function (VDF) featuring in the GPBE by using a quadrature formula. The methodsare very ﬂexible: the number of nodes in the quadrature corresponds to the number of disperse phasessimulated. The more the nodes, the better the quadrature approximation; more nodes, however, entailalso more complexity and more computational effort. For monovariate systems,
i
.
e
., systems withonly one internal coordinate in the generalized sense, the methods are entirely equivalent from atheoretical standpoint; computationally, however, they differ substantially.To conclude my PhD, I used DQMOM to simulate the dynamics of two polydisperse powdersinitially arranged as two superposed, segregated packed systems (Mazzei, 2008). As ﬂuidization
occurs, the simulation tracks the evolution in time and physical space of the quadrature nodes andweights and predicts the degree of mixing attained by the system. To validate the method, I comparedcomputational predictions with experimental results.
References
Foscolo, P. U. & Gibilaro, L. G. 1987. Fluid dynamic stability of ﬂuidized suspensions: The particlebed model. Chem. Eng. Sci.
42
, 1489.
Luca Mazzei
References
3Mazzei, L., Lettieri, P., Elson, T. & Colman, D. 2006. A revised monodimensional particle bed modelfor ﬂuidized beds. Chem. Eng. Sci.
61
, 1958.Mazzei, L. & Lettieri, P. 2007. A drag force closure for uniformly-dispersed ﬂuidized suspensions.Chem. Eng. Sci.
62
, 6129.Mazzei, L. & Lettieri, P. 2008. CFD investigation into the dynamics and stability of homogeneousﬂuidized suspensions. Chem. Eng. Sci. Accepted for publication.Mazzei, L. 2008. Eulerian modeling and computational ﬂuid dynamics simulation of mono and poly-disperse ﬂuidized suspensions. Ph.D. Dissertation. Department of Chemical Engineering, Univer-sity College London.Owoyemi, O., Mazzei, L. & Lettieri, P. 2007. CFD modeling of binary-ﬂuidized suspensions andinvestigation of role of particle-particle drag on mixing and segregation. AIChE J.
53
, 1924.Richardson, J. F. & Zaki, W. N. 1954. Sedimentation and ﬂuidization: Part I. Trans. Inst. Chem. Eng.
32
, 35.

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