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E-raamat: Computational Models for Polydisperse Particulate and Multiphase Systems

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(Iowa State University), (Politecnico di Torino)
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Computational Models for Polydisperse Particulate and Multiphase SystemsSuurem pilt
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Providing a clear description of the theory of polydisperse multiphase flows, with emphasis on the mesoscale modelling approach and its relationship with microscale and macroscale models, this all-inclusive introduction is ideal whether you are working in industry or academia. Theory is linked to practice through discussions of key real-world cases (particle/droplet/bubble coalescence, break-up, nucleation, advection and diffusion and physical- and phase-space), providing valuable experience in simulating systems that can be applied to your own applications. Practical cases of QMOM, DQMOM, CQMOM, EQMOM and ECQMOM are also discussed and compared, as are realizable finite-volume methods. This provides the tools you need to use quadrature-based moment methods, choose from the many available options, and design high-order numerical methods that guarantee realizable moment sets. In addition to the numerous practical examples, MATLAB scripts for several algorithms are also provided, so you can apply the methods described to practical problems straight away.

With this all-inclusive introduction to polydisperse multiphase flows, you will learn how to use quadrature-based moment methods and design high-order numerical methods that guarantee realizable moment sets. Theory is linked to practice through numerous real-world examples, whilst MATLAB® scripts are provided to apply the methods to your own practical problems.

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All-inclusive introduction to polydisperse multiphase flows linking theory to practice through numerous real-world examples and MATLAB® scripts for key algorithms.
Preface xiii
Notation xvii
1 Introduction
1(29)
1.1 Disperse multiphase flows
1(2)
1.2 Two example systems
3(11)
1.2.1 The population-balance equation for fine particles
3(5)
1.2.2 The kinetic equation for gas--particle flow
8(6)
1.3 The mesoscale modeling approach
14(9)
1.3.1 Relation to microscale models
16(2)
1.3.2 Number-density functions
18(1)
1.3.3 The kinetic equation for the disperse phase
19(1)
1.3.4 Closure at the mesoscale level
20(1)
1.3.5 Relation to macroscale models
20(3)
1.4 Closure methods for moment-transport equations
23(4)
1.4.1 Hydrodynamic models
23(2)
1.4.2 Moment methods
25(2)
1.5 A road map to
Chapters 2-8
27(3)
2 Mesoscale description of polydisperse systems
30(17)
2.1 Number-density functions (NDF)
30(5)
2.1.1 Length-based NDF
32(1)
2.1.2 Volume-based NDF
33(1)
2.1.3 Mass-based NDF
33(1)
2.1.4 Velocity-based NDF
34(1)
2.2 The NDF transport equation
35(3)
2.2.1 The population-balance equation (PBE)
35(2)
2.2.2 The generalized population-balance equation (GPBE)
37(1)
2.2.3 The closure problem
37(1)
2.3 Moment-transport equations
38(5)
2.3.1 Moment-transport equations for a PBE
38(2)
2.3.2 Moment-transport equations for a GPBE
40(3)
2.4 Flow regimes for the PBE
43(2)
2.4.1 Laminar PBE
43(1)
2.4.2 Turbulent PBE
44(1)
2.5 The moment-closure problem
45(2)
3 Quadrature-based moment methods
47(55)
3.1 Univariate distributions
47(15)
3.1.1 Gaussian quadrature
49(2)
3.1.2 The product-difference (PD) algorithm
51(2)
3.1.3 The Wheeler algorithm
53(2)
3.1.4 Consistency of a moment set
55(7)
3.2 Multivariate distributions
62(20)
3.2.1 Brute-force QMOM
63(5)
3.2.2 Tensor-product QMOM
68(6)
3.2.3 Conditional QMOM
74(8)
3.3 The extended quadrature method of moments (EQMOM)
82(17)
3.3.1 Relationship to orthogonal polynomials
83(1)
3.3.2 Univariate EQMOM
84(7)
3.3.3 Evaluation of integrals with the EQMOM
91(2)
3.3.4 Multivariate EQMOM
93(6)
3.4 The direct quadrature method of moments (DQMOM)
99(3)
4 The generalized population-balance equation
102(34)
4.1 Particle-based definition of the NDF
102(8)
4.1.1 Definition of the NDF for granular systems
102(3)
4.1.2 NDF estimation methods
105(2)
4.1.3 Definition of the NDF for fluid-particle systems
107(3)
4.2 From the multi-particle--fluid joint PDF to the GPBE
110(4)
4.2.1 The transport equation for the multi-particle joint PDF
111(1)
4.2.2 The transport equation for the single-particle joint PDF
112(1)
4.2.3 The transport equation for the NDF
112(1)
4.2.4 The closure problem
113(1)
4.3 Moment-transport equations
114(16)
4.3.1 A few words about phase-space integration
114(2)
4.3.2 Disperse-phase number transport
116(1)
4.3.3 Disperse-phase volume transport
116(1)
4.3.4 Fluid-phase volume transport
117(1)
4.3.5 Disperse-phase mass transport
118(3)
4.3.6 Fluid-phase mass transport
121(2)
4.3.7 Disperse-phase momentum transport
123(1)
4.3.8 Fluid-phase momentum transport
124(3)
4.3.9 Higher-order moment transport
127(3)
4.4 Moment closures for the GPBE
130(6)
5 Mesoscale models for physical and chemical processes
136(78)
5.1 An overview of mesoscale modeling
136(11)
5.1.1 Mesoscale models in the GPBE
137(4)
5.1.2 Formulation of mesoscale models
141(4)
5.1.3 Relation to macroscale models
145(2)
5.2 Phase-space advection: mass and heat transfer
147(14)
5.2.1 Mesoscale variables for particle size
149(3)
5.2.2 Size change for crystalline and amorphous particles
152(3)
5.2.3 Non-isothermal systems
155(1)
5.2.4 Mass transfer to gas bubbles
156(2)
5.2.5 Heat/mass transfer to liquid droplets
158(2)
5.2.6 Momentum change due to mass transfer
160(1)
5.3 Phase-space advection: momentum transfer
161(16)
5.3.1 Buoyancy and drag forces
162(9)
5.3.2 Virtual-mass and lift forces
171(2)
5.3.3 Boussinesq--Basset, Brownian, and thermophoretic forces
173(2)
5.3.4 Final expressions for the mesoscale acceleration models
175(2)
5.4 Real-space advection
177(6)
5.4.1 The pseudo-homogeneous or dusty-gas model
179(1)
5.4.2 The equilibrium or algebraic Eulerian model
180(1)
5.4.3 The Eulerian two-fluid model
181(1)
5.4.4 Guidelines for real-space advection
182(1)
5.5 Diffusion processes
183(6)
5.5.1 Phase-space diffusion
184(3)
5.5.2 Physical-space diffusion
187(1)
5.5.3 Mixed phase-and physical-space diffusion
188(1)
5.6 Zeroth-order point processes
189(3)
5.6.1 Formation of the disperse phase
189(2)
5.6.2 Nucleation of crystals from solution
191(1)
5.6.3 Nucleation of vapor bubbles in a boiling liquid
191(1)
5.7 First-order point processes
192(10)
5.7.1 Particle filtration and deposition
193(2)
5.7.2 Particle breakage
195(7)
5.8 Second-order point processes
202(12)
5.8.1 Derivation of the source term
203(2)
5.8.2 Source terms for aggregation and coalescence
205(1)
5.8.3 Aggregation kernels for fine particles
206(6)
5.8.4 Coalescence kernels for droplets and bubbles
212(2)
6 Hard-sphere collision models
214(52)
6.1 Monodisperse hard-sphere collisions
215(21)
6.1.1 The Boltzmann collision model
217(1)
6.1.2 The collision term for arbitrary moments
218(3)
6.1.3 Collision angles and the transformation matrix
221(2)
6.1.4 Integrals over collision angles
223(7)
6.1.5 The collision term for integer moments
230(6)
6.2 Polydisperse hard-sphere collisions
236(10)
6.2.1 Collision terms for arbitrary moments
237(5)
6.2.2 The third integral over collision angles
242(1)
6.2.3 Collision terms for integer moments
243(3)
6.3 Kinetic models
246(4)
6.3.1 Monodisperse particles
246(2)
6.3.2 Polydisperse particles
248(2)
6.4 Moment-transport equations
250(11)
6.4.1 Monodisperse particles
251(4)
6.4.2 Polydisperse particles
255(6)
6.5 Application of quadrature to collision terms
261(5)
6.5.1 Flux terms
261(2)
6.5.2 Source terms
263(3)
7 Solution methods for homogeneous systems
266(63)
7.1 Overview of methods
266(3)
7.2 Class and sectional methods
269(20)
7.2.1 Univariate PBE
269(10)
7.2.2 Bivariate and multivariate PBE
279(4)
7.2.3 Collisional KE
283(6)
7.3 The method of moments
289(11)
7.3.1 Univariate PBE
290(6)
7.3.2 Bivariate and multivariate PBE
296(1)
7.3.3 Collisional KE
297(3)
7.4 Quadrature-based moment methods
300(15)
7.4.1 Univariate PBE
301(6)
7.4.2 Bivariate and multivariate PBE
307(7)
7.4.3 Collisional KE
314(1)
7.5 Monte Carlo methods
315(4)
7.6 Example homogeneous PBE
319(10)
7.6.1 A few words on the spatially homogeneous PBE
319(4)
7.6.2 Comparison between the QMOM and the DQMOM
323(1)
7.6.3 Comparison between the CQMOM and Monte Carlo
324(5)
8 Moment methods for inhomogeneous systems
329(74)
8.1 Overview of spatial modeling issues
329(11)
8.1.1 Realizability
330(2)
8.1.2 Particle trajectory crossing
332(3)
8.1.3 Coupling between active and passive internal coordinates
335(2)
8.1.4 The QMOM versus the DQMOM
337(3)
8.2 Kinetics-based finite-volume methods
340(9)
8.2.1 Application to PBE
341(4)
8.2.2 Application to KE
345(2)
8.2.3 Application to GPBE
347(2)
8.3 Inhomogeneous PBE
349(13)
8.3.1 Moment-transport equations
349(1)
8.3.2 Standard finite-volume schemes for moments
350(3)
8.3.3 Realizable finite-volume schemes for moments
353(5)
8.3.4 Example results for an inhomogeneous PBE
358(4)
8.4 Inhomogeneous KE
362(11)
8.4.1 The moment-transport equation
363(1)
8.4.2 Operator splitting for moment equations
363(1)
8.4.3 A realizable finite-volume scheme for bivariate velocity moments
364(2)
8.4.4 Example results for an inhomogeneous KE
366(7)
8.5 Inhomogeneous GPBE
373(28)
8.5.1 Classes of GPBE
373(3)
8.5.2 Spatial transport with known scalar-dependent velocity
376(1)
8.5.3 Example results with known scalar-dependent velocity
377(4)
8.5.4 Spatial transport with scalar-conditioned velocity
381(7)
8.5.5 Example results with scalar-conditioned velocity
388(8)
8.5.6 Spatial transport of the velocity-scalar NDF
396(5)
8.6 Concluding remarks
401(2)
Appendix A Moment-inversion algorithms
403(18)
A.1 Univariate quadrature
403(2)
A.1.1 The PD algorithm
403(1)
A.1.2 The adaptive Wheeler algorithm
404(1)
A.2 Moment-correction algorithms
405(3)
A.2.1 The correction algorithm of McGraw
405(2)
A.2.2 The correction algorithm of Wright
407(1)
A.3 Multivariate quadrature
408(5)
A.3.1 Brute-force QMOM
408(2)
A.3.2 Tensor-product QMOM
410(2)
A.3.3 The CQMOM
412(1)
A.4 The EQMOM
413(8)
A.4.1 Beta EQMOM
413(3)
A.4.2 Gamma EQMOM
416(2)
A.4.3 Gaussian EQMOM
418(3)
Appendix B Kinetics-based finite-volume methods
421(20)
B.1 Spatial dependence of GPBE
421(2)
B.2 Realizable FVM
423(4)
B.3 Advection
427(2)
B.4 Free transport
429(5)
B.5 Mixed advection
434(3)
B.6 Diffusion
437(4)
Appendix C Moment methods with hyperbolic equations
441(9)
C.1 A model kinetic equation
441(1)
C.2 Analytical solution for segregated initial conditions
442(2)
C.2.1 Segregating solution
442(1)
C.2.2 Mixing solution
443(1)
C.3 Moments and the quadrature approximation
444(3)
C.3.1 Moments of segregating solution
444(2)
C.3.2 Moments of mixing solution
446(1)
C.4 Application of QBMM
447(3)
C.4.1 The moment-transport equation
447(1)
C.4.2 Transport equations for weights and abscissas
448(2)
Appendix D The direct quadrature method of moments fully conservative
450(9)
D.1 Inhomogeneous PBE
450(1)
D.2 Standard DQMOM
450(3)
D.3 DQMOM-FC
453(2)
D.4 Time integration
455(4)
References 459(29)
Index 488
Daniele L. Marchisio is an Associate Professor at the Politecnico di Torino, Italy, where he received his Ph.D. in 2001. He held visiting positions at the Laboratoire des Science du Génie Chimique, CNRSENSIC (Nancy, France), Iowa State University (USA), Eidgenössische Technische Hochschule Zürich (Switzerland), University College London (UK) and has been an invited professor at Aalborg University (Denmark) and University of Valladolid (Spain). He acts as referee for the key international journals of his field of research. He has authored 60 scientific papers, five book chapters and co-edited the volume Multiphase Reacting Flows (2007). Rodney O. Fox is the Anson Marston Distinguished Professor of Engineering at Iowa State University, Associate Scientist at the US-DOE Ames Laboratory and Senior Research Fellow in the EM2C laboratory at the Ecole Centrale Paris, France. His numerous professional awards include a NSF Presidential Young Investigator Award in 1992 and Fellow of the American Physical Society in 2007. The impact of Fox's work touches every technological area dealing with multiphase flow and chemical reactions. His monograph Computational Models for Turbulent Reacting Flows (Cambridge University Press, 2003) offers an authoritative treatment of the field.
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