CHAPMAN, S. & COWLING, T.G. 1990 *The Mathematical Theory of Non-Uniform Gases: An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases*. Cambridge University Press.

CHEN, X. & WANG, J. 2014 A comparison of two-fluid model, dense discrete particle model and CFD-DEM method for modeling impinging gas-solid flows. *Powder Technol.* **254**, 94–102.

CHORIN, A.J. 1968 Numerical solution of the Navier-Stokes equations. *Maths Comput.* **22** (104), 745–762.

COLIN, O. & RUDGYARD, M. 2000 Development of high-order Taylor-Galerkin schemes for LES. *J. Comput. Phys.* **162** (2), 338–371.

CUNDALL, P.A. & STRACK, O.D.L. 1979 A discrete numerical model for granular assemblies. *Gèotechnique* **29** (1), 47–65.

DARELIUS, A., RASMUSON, A., VAN WACHEM, B., BJÖRN, I.N. & FOLESTAD, S. 2008 CFD simulation of the high shear mixing process using kinetic theory of granular flow and frictional stress models. *Chem. Engng Sci.* **63** (8), 2188–2197.

DENG, Y., ANSART, R., BAEYENS, J. & ZHANG, H. 2019 Flue gas desulphurization in circulating fluidized beds. *Energies* **12** (20), 3908.

DUFRESNE, Y., MOUREAU, V., LARTIGUE, G. & SIMONIN, O. 2020 A massively parallel CFD/DEM approach for reactive gas-solid flows in complex geometries using unstructured meshes. *Comput. Fluids* **198**, 104402.

FEDE, P., MOULA, G., INGRAM, A., DUMAS, T. & SIMONIN, O. 2009 3D numerical simulation and PEPT experimental investigation of pressurized gas-solid fluidized bed hydrodynamic. In *Fluids Engineering Division Summer Meeting*, vol. 43727, pp. 1833–1842. ASME.

FEDE, P. & SIMONIN, O. 2018 Direct simulation Monte Carlo predictions of coarse elastic particle statistics in fully developed turbulent channel flows: comparison with deterministic discrete particle simulation results and moment closure assumptions. *Intl J. Multiphase Flow* **108**, 25–41.

FEDE, P., SIMONIN, O. & INGRAM, A. 2016 3D numerical simulation of a lab-scale pressurized dense fluidized bed focussing on the effect of the particle-particle restitution coefficient and particle-wall boundary conditions. *Chem. Engng Sci.* **142**, 215–235.

FÉVRIER, P., SIMONIN, O. & SQUIRES, K.D. 2005 Partitioning of particle velocities in gas-solid turbulent flows into a continuous field and a spatially uncorrelated random distribution: theoretical formalism and numerical study. *J. Fluid Mech.* **533**, 1–46.

GOBIN, A., NEAU, H., SIMONIN, O., LLINAS, J.-R., REILING, V. & SÉLO, J.-L. 2003 Fluid dynamic numerical simulation of a gas phase polymerization reactor. *Intl J. Numer. Meth. Fluids* **43** (10–11), 1199–1220.

GU, Y., OZEL, A., KOLEHMAINEN, J. & SUNDARESAN, S. 2019 Computationally generated constitutive models for particle phase rheology in gas-fluidized suspensions. *J. Fluid Mech.* **860**, 318–349.

VAN DER HOEF, M.A., YE, M., VAN SINT ANNALAND, M., ANDREWS, A.T., SUNDARESAN, S. & KUIPERS, J.A.M. 2006 Multiscale modeling of gas-fluidized beds. *Adv. Chem. Engng* **31**, 65–149.

HUI, K., HAFF, P.K., UNGAR, J.E. & JACKSON, R. 1984 Boundary conditions for high-shear grain flows. *J. Fluid Mech.* **145**, 223–233.

JENKINS, J.T. 1992 Boundary conditions for rapid granular flow: flat, frictional walls. *J. Appl. Mech.* **59** (1), 120.

JENKINS, J.T. & LOUGE, M.Y. 1997 On the flux of fluctuation energy in a collisional grain flow at a flat, frictional wall. *Phys. Fluids* **9** (10), 2835–2840.

JENKINS, J.T. & SAVAGE, S.B. 1983 A theory for the rapid flow of identical, smooth, nearly elastic, spherical particles. *J. Fluid Mech.* **130**, 187–202.

JOHNSON, P.C. & JACKSON, R. 1987 Frictional–collisional constitutive relations for granular materials, with application to plane shearing. *J. Fluid Mech.* **176**, 67–93.

JOHNSON, P.C., NOTT, P. & JACKSON, R. 1990 Frictional–collisional equations of motion for participate flows and their application to chutes. *J. Fluid Mech.* **210**, 501–535.

KOMIWES, V., MEGE, P., MEIMON, Y. & HERRMANN, H. 2006 Simulation of granular flow in a fluid applied to sedimentation. *Granul. Matt.* **8**, 41–54.

KONAN, N.A., SIMONIN, O. & SQUIRES, K.D. 2006 Rough wall boundary condition derivation for particle continuum equations: validation from LES/DPS of gas-solid turbulent channel flow. In *Fluids Engineering Division Summer Meeting*, vol. 47500, pp. 1723–1732. ASME.

KRAUSHAAR, M. 2011 Application of the compressible and low-Mach number approaches to large-eddy simulation of turbulent flows in aero-engines. PhD thesis, Université de Toulouse, France.

LI, T. & BENYAHIA, S. 2012 Revisiting Johnson and Jackson boundary conditions for granular flows. *AIChE J.* **58** (7), 2058–2068.

LI, T., GRACE, J. & BI, X. 2010 Study of wall boundary condition in numerical simulations of bubbling fluidized beds. *Powder Technol.* **203** (3), 447–457.
CHAPMAN, S. & COWLING, T.G. 1990 *The Mathematical Theory of Non-Uniform Gases: An Account of the Kinetic Theory of Viscosity, Thermal Conduction and Diffusion in Gases*. Cambridge University Press.

CHEN, X. & WANG, J. 2014 A comparison of two-fluid model, dense discrete particle model and CFD-DEM method for modeling impinging gas-solid flows. *Powder Technol.* **254**, 94–102.

CHORIN, A.J. 1968 Numerical solution of the Navier-Stokes equations. *Maths Comput.* **22** (104), 745–762.

COLIN, O. & RUDGYARD, M. 2000 Development of high-order Taylor-Galerkin schemes for LES. *J. Comput. Phys.* **162** (2), 338–371.

CUNDALL, P.A. & STRACK, O.D.L. 1979 A discrete numerical model for granular assemblies. *Gèotechnique* **29** (1), 47–65.

DARELIUS, A., RASMUSON, A., VAN WACHEM, B., BJÖRN, I.N. & FOLESTAD, S. 2008 CFD simulation of the high shear mixing process using kinetic theory of granular flow and frictional stress models. *Chem. Engng Sci.* **63** (8), 2188–2197.

DENG, Y., ANSART, R., BAEYENS, J. & ZHANG, H. 2019 Flue gas desulphurization in circulating fluidized beds. *Energies* **12** (20), 3908.

DUFRESNE, Y., MOUREAU, V., LARTIGUE, G. & SIMONIN, O. 2020 A massively parallel CFD/DEM approach for reactive gas-solid flows in complex geometries using unstructured meshes. *Comput. Fluids* **198**, 104402.

FEDE, P., MOULA, G., INGRAM, A., DUMAS, T. & SIMONIN, O. 2009 3D numerical simulation and PEPT experimental investigation of pressurized gas-solid fluidized bed hydrodynamic. In *Fluids Engineering Division Summer Meeting*, vol. 43727, pp. 1833–1842. ASME.

FEDE, P. & SIMONIN, O. 2018 Direct simulation Monte Carlo predictions of coarse elastic particle statistics in fully developed turbulent channel flows: comparison with deterministic discrete particle simulation results and moment closure assumptions. *Intl J. Multiphase Flow* **108**, 25–41.

FEDE, P., SIMONIN, O. & INGRAM, A. 2016 3D numerical simulation of a lab-scale pressurized dense fluidized bed focussing on the effect of the particle-particle restitution coefficient and particle-wall boundary conditions. *Chem. Engng Sci.* **142**, 215–235.

FÉVRIER, P., SIMONIN, O. & SQUIRES, K.D. 2005 Partitioning of particle velocities in gas-solid turbulent flows into a continuous field and a spatially uncorrelated random distribution: theoretical formalism and numerical study. *J. Fluid Mech.* **533**, 1–46.

GOBIN, A., NEAU, H., SIMONIN, O., LLINAS, J.-R., REILING, V. & SÉLO, J.-L. 2003 Fluid dynamic numerical simulation of a gas phase polymerization reactor. *Intl J. Numer. Meth. Fluids* **43** (10–11), 1199–1220.

GU, Y., OZEL, A., KOLEHMAINEN, J. & SUNDARESAN, S. 2019 Computationally generated constitutive models for particle phase rheology in gas-fluidized suspensions. *J. Fluid Mech.* **860**, 318–349.

VAN DER HOEF, M.A., YE, M., VAN SINT ANNALAND, M., ANDREWS, A.T., SUNDARESAN, S. & KUIPERS, J.A.M. 2006 Multiscale modeling of gas-fluidized beds. *Adv. Chem. Engng* **31**, 65–149.

HUI, K., HAFF, P.K., UNGAR, J.E. & JACKSON, R. 1984 Boundary conditions for high-shear grain flows. *J. Fluid Mech.* **145**, 223–233.

JENKINS, J.T. 1992 Boundary conditions for rapid granular flow: flat, frictional walls. *J. Appl. Mech.* **59** (1), 120.

JENKINS, J.T. & LOUGE, M.Y. 1997 On the flux of fluctuation energy in a collisional grain flow at a flat, frictional wall. *Phys. Fluids* **9** (10), 2835–2840.

JENKINS, J.T. & SAVAGE, S.B. 1983 A theory for the rapid flow of identical, smooth, nearly elastic, spherical particles. *J. Fluid Mech.* **130**, 187–202.

JOHNSON, P.C. & JACKSON, R. 1987 Frictional–collisional constitutive relations for granular materials, with application to plane shearing. *J. Fluid Mech.* **176**, 67–93.

JOHNSON, P.C., NOTT, P. & JACKSON, R. 1990 Frictional–collisional equations of motion for participate flows and their application to chutes. *J. Fluid Mech.* **210**, 501–535.

KOMIWES, V., MEGE, P., MEIMON, Y. & HERRMANN, H. 2006 Simulation of granular flow in a fluid applied to sedimentation. *Granul. Matt.* **8**, 41–54.

KONAN, N.A., SIMONIN, O. & SQUIRES, K.D. 2006 Rough wall boundary condition derivation for particle continuum equations: validation from LES/DPS of gas-solid turbulent channel flow. In *Fluids Engineering Division Summer Meeting*, vol. 47500, pp. 1723–1732. ASME.

KRAUSHAAR, M. 2011 Application of the compressible and low-Mach number approaches to large-eddy simulation of turbulent flows in aero-engines. PhD thesis, Université de Toulouse, France.

LI, T. & BENYAHIA, S. 2012 Revisiting Johnson and Jackson boundary conditions for granular flows. *AIChE J.* **58** (7), 2058–2068.

LI, T., GRACE, J. & BI, X. 2010 Study of wall boundary condition in numerical simulations of bubbling fluidized beds. *Powder Technol.* **203** (3), 447–457.
Investigation of near-wall particle statistics

LOUGE, M. Y. 1994 Computer simulations of rapid granular flows of spheres interacting with a flat, frictional boundary. *Phys. Fluids* **6** (7), 2253–2269.

LUBACHEVSKY, B.D. 1991 How to simulate billiards and similar systems. *J. Comput. Phys.* **94** (2), 255–283.

LUN, C.K.K. & SAVAGE, S.B. 1986 The effects of an impact velocity dependent coefficient of restitution on stresses developed by sheared granular materials. *Acta Mech.* **63** (1), 15–44.

MÉCHITOUA, N., BOUCKER, M., LAVIÉVILLE, J., PIGNY, S. & SERRE, G. 2003 An unstructured finite volume solver for two phase water/vapour flows based on an elliptic oriented fractional step method. In *NURETH 10, Seoul, South Korea*.

MORIOKA, S. & NAKAJIMA, T. 1987 Modeling of gas and solid particles 2-phase flow and application to fluidized-bed. *J. Theor. Appl. Mech.* **6** (1), 77–88.

MOUREAU, V., DOMINGO, P. & VERVISCH, L. 2011 Design of a massively parallel CFD code for complex geometries. *C. R. Méc.* **339** (2–3), 141–148.

MUELLER, G.E. 1992 Radial void fraction distributions in randomly packed fixed beds of uniformly sized spheres in cylindrical containers. *Powder Technol.* **72** (3), 269–275.

MUELLER, G.E. 1997 Numerical simulation of packed beds with monosized spheres in cylindrical containers. *Powder Technol.* **92** (2), 179–183.

NEAU, H., PIGOU, M., FEDE, P., ANSART, R., BAUDRY, C., MÉRIGOUX, N., LAVIÉVILLE, J., FOURNIER, Y., RENON, N. & SIMONIN, O. 2020 Massively parallel numerical simulation using up to 36 000 CPU cores of an industrial-scale polydispersed reactive pressurized fluidized bed with a mesh of one billion cells. *Powder Technol.* **366**, 906–924.

NICOT, F., HADDA, N., GUESSASMA, M., FORTIN, J. & MILLET, O. 2013 On the definition of the stress tensor in granular media. *Intl J. Solids Struct.* **50** (14–15), 2508–2517.

NIGMETOVA, A. 2019 DEM-CFD simulation of dilute and dense particle laden flows: application to a lab-scale pressurized dense fluidized bed. PhD thesis, Université de Toulouse, France.

NIGMETOVA, A., MASI, E., SIMONIN, O., DUFRESNE, Y. & MOUREAU, V. 2022 Three-dimensional DEM-CFD simulation of a lab-scale fluidized bed to support the development of two-fluid model approach. *Intl J. Multiphase Flow* **156**, 104189.

OCONE, R., SUNDARESAN, S. & JACKSON, R. 1993 Gas-particle flow in a duct of arbitrary inclination with particle-particle interactions. *AIChE J.* **39** (8), 1261–1271.

POPOFF, B. & BRAUN, M. 2007 A Lagrangian approach to dense particulate flows. In *International Conference on Multiphase Flow, Leipzig, Germany* (ed. M. Sommerfield).

RADENKOVIC, D. & SIMONIN, O. 2018 Stochastic modelling of three-dimensional particle rebound from isotropic rough wall surface. *Intl J. Multiphase Flow* **109**, 35–50.

REEKS, M.W. 1991 On a kinetic equation for the transport of particles in turbulent flows. *Phys. Fluids A* **3** (3), 446–456.

ROCKE, F.A. 1971 The cylindrically ordered packing of equal spheres. *Powder Technol.* **4** (4), 180–186.

SABATIER, F., ANSART, R., ZHANG, H., BAEYENS, J. & SIMONIN, O. 2020 Experiments support simulations by the NEPTUNE_CFD code in an upflow bubbling fluidized bed reactor. *Chem. Engng J.* **385**, 123568.

SAKIZ, M. & SIMONIN, O. 1998 Continuum modelling and Lagrangian simulation of the turbulent transport of particle kinetic stresses in a vertical gas-solid channel flow. In *3rd International Conference on Multiphase Flows, Lyon, France*, vol. 66, pp. 70–91.

SAKIZ, M. & SIMONIN, O. 1999 Development and validation of continuum particle wall boundary conditions using Lagrangian simulation of a vertical gas-solid channel flow. In *Proc. ASME Fluids Engineering Division Summer Meeting, FEDSM99-7898*. ASME.

SCHAEFFER, D.G. 1987 Instability in the evolution equations describing incompressible granular flow. *J. Differ. Equ.* **66** (1), 19–50.

SCHNEIDERBAUER, S., SCHELLANDER, D., LÖDERER, A. & PIRKER, S. 2012 Non-steady state boundary conditions for collisional granular flows at flat frictional moving walls. *Intl J. Multiphase Flow* **43**, 149–156.

VON SECKENDORFF, J. & HINRICHSEN, O. 2021 Review on the structure of random packed-beds. *Can. J. Chem. Engng* **99**, S703–S733.

SHARMA, A., WANG, S., PAREEK, V., YANG, H. & ZHANG, D. 2014 CFD modeling of mixing/segregation behavior of biomass and biochar particles in a bubbling fluidized bed. *Chem. Engng Sci.* **106**, 264–274.

SHU, Z., PENG, G., WANG, J., ZHANG, N., LI, S. & LIN, W. 2014 Comparative CFD analysis of heterogeneous gas-solid flow in a countercurrent downer reactor. *Ind. Engng Chem. Res.* **53** (8), 3378–3384.

SI, P., SHI, H. & YU, X. 2019 A general frictional-collisional model for dense granular flows. *Landslides* **16**, 485–496.
LOUGE, M. Y. 1994 Computer simulations of rapid granular flows of spheres interacting with a flat, frictional boundary. *Phys. Fluids* **6** (7), 2253–2269.

LUBACHEVSKY, B.D. 1991 How to simulate billiards and similar systems. *J. Comput. Phys.* **94** (2), 255–283.

LUN, C.K.K. & SAVAGE, S.B. 1986 The effects of an impact velocity dependent coefficient of restitution on stresses developed by sheared granular materials. *Acta Mech.* **63** (1), 15–44.

MÉCHITOUA, N., BOUCKER, M., LAVIÉVILLE, J., PIGNY, S. & SERRE, G. 2003 An unstructured finite volume solver for two phase water/vapour flows based on an elliptic oriented fractional step method. In *NURETH 10, Seoul, South Korea*.

MORIOKA, S. & NAKAJIMA, T. 1987 Modeling of gas and solid particles 2-phase flow and application to fluidized-bed. *J. Theor. Appl. Mech.* **6** (1), 77–88.

MOUREAU, V., DOMINGO, P. & VERVISCH, L. 2011 Design of a massively parallel CFD code for complex geometries. *C. R. Méc.* **339** (2–3), 141–148.

MUELLER, G.E. 1992 Radial void fraction distributions in randomly packed fixed beds of uniformly sized spheres in cylindrical containers. *Powder Technol.* **72** (3), 269–275.

MUELLER, G.E. 1997 Numerical simulation of packed beds with monosized spheres in cylindrical containers. *Powder Technol.* **92** (2), 179–183.

NEAU, H., PIGOU, M., FEDE, P., ANSART, R., BAUDRY, C., MÉRIGOUX, N., LAVIÉVILLE, J., FOURNIER, Y., RENON, N. & SIMONIN, O. 2020 Massively parallel numerical simulation using up to 36 000 CPU cores of an industrial-scale polydispersed reactive pressurized fluidized bed with a mesh of one billion cells. *Powder Technol.* **366**, 906–924.

NICOT, F., HADDA, N., GUESSASMA, M., FORTIN, J. & MILLET, O. 2013 On the definition of the stress tensor in granular media. *Intl J. Solids Struct.* **50** (14–15), 2508–2517.

NIGMETOVA, A. 2019 DEM-CFD simulation of dilute and dense particle laden flows: application to a lab-scale pressurized dense fluidized bed. PhD thesis, Université de Toulouse, France.

NIGMETOVA, A., MASI, E., SIMONIN, O., DUFRESNE, Y. & MOUREAU, V. 2022 Three-dimensional DEM-CFD simulation of a lab-scale fluidized bed to support the development of two-fluid model approach. *Intl J. Multiphase Flow* **156**, 104189.

OCONE, R., SUNDARESAN, S. & JACKSON, R. 1993 Gas-particle flow in a duct of arbitrary inclination with particle-particle interactions. *AIChE J.* **39** (8), 1261–1271.

POPOFF, B. & BRAUN, M. 2007 A Lagrangian approach to dense particulate flows. In *International Conference on Multiphase Flow, Leipzig, Germany* (ed. M. Sommerfield).

RADENKOVIC, D. & SIMONIN, O. 2018 Stochastic modelling of three-dimensional particle rebound from isotropic rough wall surface. *Intl J. Multiphase Flow* **109**, 35–50.

REEKS, M.W. 1991 On a kinetic equation for the transport of particles in turbulent flows. *Phys. Fluids A* **3** (3), 446–456.

ROCKE, F.A. 1971 The cylindrically ordered packing of equal spheres. *Powder Technol.* **4** (4), 180–186.

SABATIER, F., ANSART, R., ZHANG, H., BAEYENS, J. & SIMONIN, O. 2020 Experiments support simulations by the NEPTUNE_CFD code in an upflow bubbling fluidized bed reactor. *Chem. Engng J.* **385**, 123568.

SAKIZ, M. & SIMONIN, O. 1998 Continuum modelling and Lagrangian simulation of the turbulent transport of particle kinetic stresses in a vertical gas-solid channel flow. In *3rd International Conference on Multiphase Flows, Lyon, France*, vol. 66, pp. 70–91.

SAKIZ, M. & SIMONIN, O. 1999 Development and validation of continuum particle wall boundary conditions using Lagrangian simulation of a vertical gas-solid channel flow. In *Proc. ASME Fluids Engineering Division Summer Meeting, FEDSM99-7898*. ASME.

SCHAEFFER, D.G. 1987 Instability in the evolution equations describing incompressible granular flow. *J. Differ. Equ.* **66** (1), 19–50.

SCHNEIDERBAUER, S., SCHELLANDER, D., LÖDERER, A. & PIRKER, S. 2012 Non-steady state boundary conditions for collisional granular flows at flat frictional moving walls. *Intl J. Multiphase Flow* **43**, 149–156.

VON SECKENDORFF, J. & HINRICHSEN, O. 2021 Review on the structure of random packed-beds. *Can. J. Chem. Engng* **99**, S703–S733.

SHARMA, A., WANG, S., PAREEK, V., YANG, H. & ZHANG, D. 2014 CFD modeling of mixing/segregation behavior of biomass and biochar particles in a bubbling fluidized bed. *Chem. Engng Sci.* **106**, 264–274.

SHU, Z., PENG, G., WANG, J., ZHANG, N., LI, S. & LIN, W. 2014 Comparative CFD analysis of heterogeneous gas-solid flow in a countercurrent downer reactor. *Ind. Engng Chem. Res.* **53** (8), 3378–3384.

SI, P., SHI, H. & YU, X. 2019 A general frictional-collisional model for dense granular flows. *Landslides* **16**, 485–496.
SIMONIN, O. 2000 *Statistical and Continuum Modelling of Turbulent Reactive Particulate Flows. Part I: Theoretical Derivation of Dispersed Phase Eulerian Modelling from Probability Density Function Kinetic Equation*. VKI for Fluid Dynamics, Lecture Series, vol. 6. Von Karman Institute for Fluid Dynamics.

SINCLAIR, J.L. & JACKSON, R. 1989 Gas-particle flow in a vertical pipe with particle-particle interactions. *AIChE J.* **35** (9), 1473-1486.

SOLEIMANI, A., PIRKER, S. & SCHNEIDERBAUER, S. 2015a Solid boundary condition for collisional gas-solid flows at rough walls. *Powder Technol.* **281**, 28–33.

SOLEIMANI, A., SCHNEIDERBAUER, S. & PIRKER, S. 2015b CFD study of the gas-particle flow in a horizontal duct: the impact of the solids wall boundary conditions. *Procedia Engng* **102**, 1026–1037.

SOLEIMANI, A., SCHNEIDERBAUER, S. & PIRKER, S. 2015c A comparison for different wall-boundary conditions for kinetic theory based two-fluid models. *Intl J. Multiphase Flow* **71**, 94–97.

SOMMERFELD, M. & HUBER, N. 1999 Experimental analysis and modelling of particle-wall collisions. *Intl J. Multiphase Flow* **25** (6-7), 1457–1489.

SRIVASTAVA, A. & SUNDARESAN, S. 2003 Analysis of a frictional-kinetic model for gas-particle flow. *Powder Technol.* **129** (1-3), 72-85.

THONGLIMP, V., HIQUILY, N. & LAGUERIE, C. 1984 Vitesse minimale de fluidisation et expansion des couches fluidisées par un gaz. *Powder Technol.* **38** (3), 233-253.

TINGATE, G.A. 1973 Some geometrical properties of packings of equal spheres in cylindrical vessels. *Nucl. Engng Des.* **24** (2), 153-179.

WANG, J. 2020 Continuum theory for dense gas-solid flow: a state-of-the-art review. *Chem. Engng Sci.* **215**, 115428.

WEN, C.-Y. & YU, Y.H. 1966 Mechanics of fluidization. In *Fluid Particle Technology, Chem. Eng. Progress. Symposium Series*, vol. 62, pp. 100–111. American Institute of Chemical Engineers.

WENSRICH, C.M. 2012 Boundary structure in dense random packing of monosize spherical particles. *Powder Technol.* **219**, 118–127.

WENSRICH, C.M. 2014 Stress, stress-asymmetry and contact moments in granular matter. *Granul. Matt.* **16** (4), 597–608.

ZAICHIK, L.I., OESTERLÉ, B. & ALIPCHENKOV, V.M. 2004 On the probability density function model for the transport of particles in anisotropic turbulent flow. *Phys. Fluids* **16** (6), 1956–1964.

ZHANG, W., THOMPSON, K.E., REED, A.H. & BEENKEN, L. 2006 Relationship between packing structure and porosity in fixed beds of equilateral cylindrical particles. *Chem. Engng Sci.* **61** (24), 8060–8074.

ZHAO, Y., DING, T., ZHU, L. & ZHONG, Y. 2016 A specularity coefficient model and its application to dense particulate flow simulations. *Ind. Engng Chem. Res.* **55** (5), 1439–1448.
SIMONIN, O. 2000 *Statistical and Continuum Modelling of Turbulent Reactive Particulate Flows. Part I: Theoretical Derivation of Dispersed Phase Eulerian Modelling from Probability Density Function Kinetic Equation*. VKI for Fluid Dynamics, Lecture Series, vol. 6. Von Karman Institute for Fluid Dynamics.

SINCLAIR, J.L. & JACKSON, R. 1989 Gas-particle flow in a vertical pipe with particle-particle interactions. *AIChE J.* **35** (9), 1473-1486.

SOLEIMANI, A., PIRKER, S. & SCHNEIDERBAUER, S. 2015a Solid boundary condition for collisional gas-solid flows at rough walls. *Powder Technol.* **281**, 28–33.

SOLEIMANI, A., SCHNEIDERBAUER, S. & PIRKER, S. 2015b CFD study of the gas-particle flow in a horizontal duct: the impact of the solids wall boundary conditions. *Procedia Engng* **102**, 1026–1037.

SOLEIMANI, A., SCHNEIDERBAUER, S. & PIRKER, S. 2015c A comparison for different wall-boundary conditions for kinetic theory based two-fluid models. *Intl J. Multiphase Flow* **71**, 94–97.

SOMMERFELD, M. & HUBER, N. 1999 Experimental analysis and modelling of particle-wall collisions. *Intl J. Multiphase Flow* **25** (6-7), 1457–1489.

SRIVASTAVA, A. & SUNDARESAN, S. 2003 Analysis of a frictional-kinetic model for gas-particle flow. *Powder Technol.* **129** (1-3), 72-85.

THONGLIMP, V., HIQUILY, N. & LAGUERIE, C. 1984 Vitesse minimale de fluidisation et expansion des couches fluidisées par un gaz. *Powder Technol.* **38** (3), 233-253.

TINGATE, G.A. 1973 Some geometrical properties of packings of equal spheres in cylindrical vessels. *Nucl. Engng Des.* **24** (2), 153-179.

WANG, J. 2020 Continuum theory for dense gas-solid flow: a state-of-the-art review. *Chem. Engng Sci.* **215**, 115428.

WEN, C.-Y. & YU, Y.H. 1966 Mechanics of fluidization. In *Fluid Particle Technology, Chem. Eng. Progress. Symposium Series*, vol. 62, pp. 100–111. American Institute of Chemical Engineers.

WENSRICH, C.M. 2012 Boundary structure in dense random packing of monosize spherical particles. *Powder Technol.* **219**, 118–127.

WENSRICH, C.M. 2014 Stress, stress-asymmetry and contact moments in granular matter. *Granul. Matt.* **16** (4), 597–608.

ZAICHIK, L.I., OESTERLÉ, B. & ALIPCHENKOV, V.M. 2004 On the probability density function model for the transport of particles in anisotropic turbulent flow. *Phys. Fluids* **16** (6), 1956–1964.

ZHANG, W., THOMPSON, K.E., REED, A.H. & BEENKEN, L. 2006 Relationship between packing structure and porosity in fixed beds of equilateral cylindrical particles. *Chem. Engng Sci.* **61** (24), 8060–8074.

ZHAO, Y., DING, T., ZHU, L. & ZHONG, Y. 2016 A specularity coefficient model and its application to dense particulate flow simulations. *Ind. Engng Chem. Res.* **55** (5), 1439–1448.