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Computational Models for Polydisperse Particulate and Multiphase Systems

Book Description

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.

Table of Contents

  1. Cover
  2. Title Page
  3. Copyright
  4. Dedication
  5. Contents
  6. Preface
  7. Notation
  8. 1. Introduction
    1. 1.1. Disperse multiphase flows
    2. 1.2. Two example systems
      1. 1.2.1. The population-balance equation for fine particles
      2. 1.2.2. The kinetic equation for gas–particle flow
    3. 1.3. The mesoscale modeling approach
      1. 1.3.1. Relation to microscale models
      2. 1.3.2. Number-density functions
      3. 1.3.3. The kinetic equation for the disperse phase
      4. 1.3.4. Closure at the mesoscale level
      5. 1.3.5. Relation to macroscale models
    4. 1.4. Closure methods for moment-transport equations
      1. 1.4.1. Hydrodynamic models
      2. 1.4.2. Moment methods
    5. 1.5. A road map to Chapters 2–8
  9. 2. Mesoscale description of polydisperse systems
    1. 2.1. Number-density functions (NDF)
      1. 2.1.1. Length-based NDF
      2. 2.1.2. Volume-based NDF
      3. 2.1.3. Mass-based NDF
      4. 2.1.4. Velocity-based NDF
    2. 2.2. The NDF transport equation
      1. 2.2.1. The population-balance equation (PBE)
      2. 2.2.2. The generalized population-balance equation (GPBE)
      3. 2.2.3. The closure problem
    3. 2.3. Moment-transport equations
      1. 2.3.1. Moment-transport equations for a PBE
      2. 2.3.2. Moment-transport equations for a GPBE
    4. 2.4. Flow regimes for the PBE
      1. 2.4.1. Laminar PBE
      2. 2.4.2. Turbulent PBE
    5. 2.5. The moment-closure problem
  10. 3. Quadrature-based moment methods
    1. 3.1. Univariate distributions
      1. 3.1.1. Gaussian quadrature
      2. 3.1.2. The product–difference (PD) algorithm
      3. 3.1.3. The Wheeler algorithm
      4. 3.1.4. Consistency of a moment set
    2. 3.2. Multivariate distributions
      1. 3.2.1. Brute-force QMOM
      2. 3.2.2. Tensor-product QMOM
      3. 3.2.3. Conditional QMOM
    3. 3.3. The extended quadrature method of moments (EQMOM)
      1. 3.3.1. Relationship to orthogonal polynomials
      2. 3.3.2. Univariate EQMOM
      3. 3.3.3. Evaluation of integrals with the EQMOM
      4. 3.3.4. Multivariate EQMOM
    4. 3.4. The direct quadrature method of moments (DQMOM)
  11. 4. The generalized population-balance equation
    1. 4.1. Particle-based definition of the NDF
      1. 4.1.1. Definition of the NDF for granular systems
      2. 4.1.2. NDF estimation methods
      3. 4.1.3. Definition of the NDF for fluid–particle systems
    2. 4.2. From the multi-particle–fluid joint PDF to the GPBE
      1. 4.2.1. The transport equation for the multi-particle joint PDF
      2. 4.2.2. The transport equation for the single-particle joint PDF
      3. 4.2.3. The transport equation for the NDF
      4. 4.2.4. The closure problem
    3. 4.3. Moment-transport equations
      1. 4.3.1. A few words about phase-space integration
      2. 4.3.2. Disperse-phase number transport
      3. 4.3.3. Disperse-phase volume transport
      4. 4.3.4. Fluid-phase volume transport
      5. 4.3.5. Disperse-phase mass transport
      6. 4.3.6. Fluid-phase mass transport
      7. 4.3.7. Disperse-phase momentum transport
      8. 4.3.8. Fluid-phase momentum transport
      9. 4.3.9. Higher-order moment transport
    4. 4.4. Moment closures for the GPBE
  12. 5. Mesoscale models for physical and chemical processes
    1. 5.1. An overview of mesoscale modeling
      1. 5.1.1. Mesoscale models in the GPBE
      2. 5.1.2. Formulation of mesoscale models
      3. 5.1.3. Relation to macroscale models
    2. 5.2. Phase-space advection: mass and heat transfer
      1. 5.2.1. Mesoscale variables for particle size
      2. 5.2.2. Size change for crystalline and amorphous particles
      3. 5.2.3. Non-isothermal systems
      4. 5.2.4. Mass transfer to gas bubbles
      5. 5.2.5. Heat/mass transfer to liquid droplets
      6. 5.2.6. Momentum change due to mass transfer
    3. 5.3. Phase-space advection: momentum transfer
      1. 5.3.1. Buoyancy and drag forces
      2. 5.3.2. Virtual-mass and lift forces
      3. 5.3.3. Boussinesq–Basset, Brownian, and thermophoretic forces
      4. 5.3.4. Final expressions for the mesoscale acceleration models
    4. 5.4. Real-space advection
      1. 5.4.1. The pseudo-homogeneous or dusty-gas model
      2. 5.4.2. The equilibrium or algebraic Eulerian model
      3. 5.4.3. The Eulerian two-fluid model
      4. 5.4.4. Guidelines for real-space advection
    5. 5.5. Diffusion processes
      1. 5.5.1. Phase-space diffusion
      2. 5.5.2. Physical-space diffusion
      3. 5.5.3. Mixed phase- and physical-space diffusion
    6. 5.6. Zeroth-order point processes
      1. 5.6.1. Formation of the disperse phase
      2. 5.6.2. Nucleation of crystals from solution
      3. 5.6.3. Nucleation of vapor bubbles in a boiling liquid
    7. 5.7. First-order point processes
      1. 5.7.1. Particle filtration and deposition
      2. 5.7.2. Particle breakage
    8. 5.8. Second-order point processes
      1. 5.8.1. Derivation of the source term
      2. 5.8.2. Source terms for aggregation and coalescence
      3. 5.8.3. Aggregation kernels for fine particles
      4. 5.8.4. Coalescence kernels for droplets and bubbles
  13. 6. Hard-sphere collision models
    1. 6.1. Monodisperse hard-sphere collisions
      1. 6.1.1. The Boltzmann collision model
      2. 6.1.2. The collision term for arbitrary moments
      3. 6.1.3. Collision angles and the transformation matrix
      4. 6.1.4. Integrals over collision angles
      5. 6.1.5. The collision term for integer moments
    2. 6.2. Polydisperse hard-sphere collisions
      1. 6.2.1. Collision terms for arbitrary moments
      2. 6.2.2. The third integral over collision angles
      3. 6.2.3. Collision terms for integer moments
    3. 6.3. Kinetic models
      1. 6.3.1. Monodisperse particles
      2. 6.3.2. Polydisperse particles
    4. 6.4. Moment-transport equations
      1. 6.4.1. Monodisperse particles
      2. 6.4.2. Polydisperse particles
    5. 6.5. Application of quadrature to collision terms
      1. 6.5.1. Flux terms
      2. 6.5.2. Source terms
  14. 7. Solution methods for homogeneous systems
    1. 7.1. Overview of methods
    2. 7.2. Class and sectional methods
      1. 7.2.1. Univariate PBE
      2. 7.2.2. Bivariate and multivariate PBE
      3. 7.2.3. Collisional KE
    3. 7.3. The method of moments
      1. 7.3.1. Univariate PBE
      2. 7.3.2. Bivariate and multivariate PBE
      3. 7.3.3. Collisional KE
    4. 7.4. Quadrature-based moment methods
      1. 7.4.1. Univariate PBE
      2. 7.4.2. Bivariate and multivariate PBE
      3. 7.4.3. Collisional KE
    5. 7.5. Monte Carlo methods
    6. 7.6. Example homogeneous PBE
      1. 7.6.1. A few words on the spatially homogeneous PBE
      2. 7.6.2. Comparison between the QMOM and the DQMOM
      3. 7.6.3. Comparison between the CQMOM and Monte Carlo
  15. 8. Moment methods for inhomogeneous systems
    1. 8.1. Overview of spatial modeling issues
      1. 8.1.1. Realizability
      2. 8.1.2. Particle trajectory crossing
      3. 8.1.3. Coupling between active and passive internal coordinates
      4. 8.1.4. The QMOM versus the DQMOM
    2. 8.2. Kinetics-based finite-volume methods
      1. 8.2.1. Application to PBE
      2. 8.2.2. Application to KE
      3. 8.2.3. Application to GPBE
    3. 8.3. Inhomogeneous PBE
      1. 8.3.1. Moment-transport equations
      2. 8.3.2. Standard finite-volume schemes for moments
      3. 8.3.3. Realizable finite-volume schemes for moments
      4. 8.3.4. Example results for an inhomogeneous PBE
    4. 8.4. Inhomogeneous KE
      1. 8.4.1. The moment-transport equation
      2. 8.4.2. Operator splitting for moment equations
      3. 8.4.3. A realizable finite-volume scheme for bivariate velocity moments
      4. 8.4.4. Example results for an inhomogeneous KE
    5. 8.5. Inhomogeneous GPBE
      1. 8.5.1. Classes of GPBE
      2. 8.5.2. Spatial transport with known scalar-dependent velocity
      3. 8.5.3. Example results with known scalar-dependent velocity
      4. 8.5.4. Spatial transport with scalar-conditioned velocity
      5. 8.5.5. Example results with scalar-conditioned velocity
      6. 8.5.6. Spatial transport of the velocity-scalar NDF
    6. 8.6. Concluding remarks
  16. Appendix A. Moment-inversion algorithms
    1. A.1. Univariate quadrature
      1. A.1.1. The PD algorithm
      2. A.1.2. The adaptive Wheeler algorithm
    2. A.2. Moment-correction algorithms
      1. A.2.1. The correction algorithm of McGraw
      2. A.2.2. The correction algorithm of Wright
    3. A.3. Multivariate quadrature
      1. A.3.1. Brute-force QMOM
      2. A.3.2. Tensor-product QMOM
      3. A.3.3. The CQMOM
    4. A.4. The EQMOM
      1. A.4.1. Beta EQMOM
      2. A.4.2. Gamma EQMOM
      3. A.4.3. Gaussian EQMOM
  17. Appendix B. Kinetics-based finite-volume methods
    1. B.1. Spatial dependence of GPBE
    2. B.2. Realizable FVM
    3. B.3. Advection
    4. B.4. Free transport
    5. B.5. Mixed advection
    6. B.6. Diffusion
  18. Appendix C. Moment methods with hyperbolic equations
    1. C.1. A model kinetic equation
    2. C.2. Analytical solution for segregated initial conditions
      1. C.2.1. Segregating solution
      2. C.2.2. Mixing solution
    3. C.3. Moments and the quadrature approximation
      1. C.3.1. Moments of segregating solution
      2. C.3.2. Moments of mixing solution
    4. C.4. Application of QBMM
      1. C.4.1. The moment-transport equation
      2. C.4.2. Transport equations for weights and abscissas
  19. Appendix D. The direct quadrature method of moments fully conservative
    1. D.1. Inhomogeneous PBE
    2. D.2. Standard DQMOM
    3. D.3. DQMOM-FC
    4. D.4. Time integration
  20. References
  21. Index