Abstract:Starting from coupled Boltzmann–Enskog (BE) kinetic equations for a two-particle system consisting of hard spheres, a hyperbolic two-fluid model for binary, hard-sphere mixtures is derived with separate mean velocities and energies for each phase. In addition to spatial transport, the BE kinetic equations account for particle–particle collisions, using an elastic hard-sphere collision model, and the Archimedes (buoyancy) force due to spatial gradients of the pressure in each phase, as well as other forces invo… Show more
“…R and r in the fluid phase) and forces (i.e. F fp and D fp ) that are specific to hard-sphere mixtures (Fox 2019). Formally, the 1-D models can be rewritten as…”
“…Starting from the kinetic-theory-based model derived from first principles in Fox (2019), the definition of the particle mass was extended to include the added mass moving with the velocity of the particle. This resulted in the compressible two-fluid model in table 1.…”
Section: Closing Remarksmentioning
confidence: 99%
“…The tensorial form of the fluid-mediated particle pressure in (2.16), and its appearance with the opposite sign in the fluid-phase momentum balance, is motivated as follows. In the kinetic-theory derivation of Fox (2019), it was shown that the mixture pressure tensor has the form regardless of the size ratio between the hard spheres. The Boltzmann–Enskog contribution leads to the term involving in the fluid-phase momentum balance when added mass is neglected ().…”
Section: Two-fluid Model For Compressible Flowsmentioning
confidence: 99%
“…Having previously investigated the case with (Fox 2019), in the remainder of this work we are particularly interested in the limiting cases with (i.e. the particles have zero mass) so that .…”
Section: Two-fluid Model For Compressible Flowsmentioning
“…R and r in the fluid phase) and forces (i.e. F fp and D fp ) that are specific to hard-sphere mixtures (Fox 2019). Formally, the 1-D models can be rewritten as…”
“…Starting from the kinetic-theory-based model derived from first principles in Fox (2019), the definition of the particle mass was extended to include the added mass moving with the velocity of the particle. This resulted in the compressible two-fluid model in table 1.…”
Section: Closing Remarksmentioning
confidence: 99%
“…The tensorial form of the fluid-mediated particle pressure in (2.16), and its appearance with the opposite sign in the fluid-phase momentum balance, is motivated as follows. In the kinetic-theory derivation of Fox (2019), it was shown that the mixture pressure tensor has the form regardless of the size ratio between the hard spheres. The Boltzmann–Enskog contribution leads to the term involving in the fluid-phase momentum balance when added mass is neglected ().…”
Section: Two-fluid Model For Compressible Flowsmentioning
confidence: 99%
“…Having previously investigated the case with (Fox 2019), in the remainder of this work we are particularly interested in the limiting cases with (i.e. the particles have zero mass) so that .…”
Section: Two-fluid Model For Compressible Flowsmentioning
“…With the stress, results from numerical simulations with particle-scale resolved can be further interrogated to extract physics other than traditional particle–fluid interaction forces. This stress could have an interesting implication for hyperbolicity of the averaged equations fundamentally important for multiphase flow theories (Ndjinga 2007; Fox 2012; Theofanous & Chang 2018; Fox 2019).…”
The nonequilibrium characteristic of fluidization is closely related to the mesoscale structures. To help uncover this relationship, we performed a particle‐level experiment in a bubbling fluidized bed making particle tracking velocimetry (PTV) measurements. We found a large Knudsen number not only in bubbles or over bubble interfaces, but also in the dense emulsion where various banded structures exist, indicating a pervasive violation of the local equilibrium. Analysis of these banded structures identified stretching, compression, or shearing between particles. Non‐Gaussian velocity distributions were found throughout the bed, including in the dense emulsion with a small Knudsen number. In future work, we expect that the nonequilibrium characteristic will act as a local criterion for the assessment of various models of bubbling fluidized beds.
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