Particle mixing and segregation are common processes used in many chemical engineering applications. For example, in chemical looping combustion¹, gas-solid mixing plays an important role when gas-solid reactions are present. In the fast pyrolysis of biomass², solid mixing in fluidized bed reactors is crucial where biomass mixes with sand which acts as the heat carrier. Characterizing mixing and segregation of particles that vary in size and density have been extensively studied for many decades.

Non-invasive experimental studies such as X-ray tomography-computed tomography^3,4 have been conducted for flow visualization within the interior of granular mixers. Granular mixing simulations can be done by using Discrete Element Methods (DEM)^5,6, where the motion of each particle is tracked, or granular flow can be modeled using a continuum approach that obeys Eulerian conservation equations of mass, momentum, and energy. The multiphase computational fluid dynamic (mCFD) models are multi-fluid models based on a random field statistical approach^7,8 to gas-solid flows. For systems with many particles, DEM simulations become infeasible as the computational expense increases with particles, but the Eulerian model is attractive.

A mixing index is often used to quantify solid mixing, and there are many mixing indices proposed over several decades^9–11. The most widely used index is Lacey's¹² index which gives the ratio of how much mixing has occurred to how much could occur. It is computed by measuring the departure of the sample variance in the fraction of particles of interest at a given state from the completely segregated system normalized by the difference in the fraction between completely segregated and perfectly mixed systems. There are several variants of the Lacey index, such as Kramer¹³ and Rose⁹, that use the standard deviation instead of the variance.

Most mixing indices are based on particle data obtained from either experiments or DEM simulations, and these methods correspond to one realization of particle positions. However, multiphase computational fluid dynamics of gas-solid flow are based on averaged field representations. Therefore, the characterization of mixing in a particle realization is different from characterizing it in an averaged field representation. The question of the applicability of Lacey's index to mCFD is then not straightforward. Mixing in multiphase CFD is based on average volume fraction fields, and an index must be based on continuous fields defined on a grid.

Two new mixing indices are developed and introduced for gas-solid mixing based on average solid volume fraction fields. These mixing indices are the richness index that characterizes gas-solid mixing, and the solid fraction mixing index that characterizes solid-solid mixing. The index for gas-solid mixing is based on the particle volume fraction fields, and the index for solid-solid mixing is based on solid fraction fields. In a polydisperse system of particles, the solid fraction of a particle species is defined as its volume fraction divided by the total particle volume fraction. The idea of the new index is analogous to the mixture fraction from single-phase mixing of fuel and oxidizer in combustion.

The richness index normalizes gas-solid mixing characteristics spatially by performing a local linear transformation and by eliminating dependence on the global average of the total solid volume fraction. Therefore, the richness mixing index scales the volume fraction field between 0 (pure gas) and 1 (fully packed solids), indicating the local departure from perfect mixing corresponding to 0.5. The solid fraction mixing index, , employs a similar idea. It scales the solid fraction field between 0 (absence of particle A) and 1 (only particle A is present), indicating the local departure from perfect mixing corresponding to 0.5, thereby eliminating the dependence on the global average of the solid fraction of individual solid particle types. Perfect gas-solid mixing corresponds to the local total solid volume fraction being equal to the global average total solid volume fraction. In contrast, perfect solid-solid mixing corresponds to the local solid fraction of solid type A being equal to the global average solid fraction of A. These mixing indices are fields that capture spatial mixing patterns in a device. They are, in that sense, local and are hence called local mixing indices.

The performance of the new mixing indices is tested on two fluidized bed applications. The first application is a three-dimensional simulation of the mixing of biomass and sand in a fluidized bed reactor designed for fast pyrolysis. Two designs with different biomass injection locations are evaluated to determine whether a higher or lower injection location for the biomass yields better mixing. The second application is a two-dimensional simulation of segregation of an initially perfectly mixed binary mixture of particles with similar densities but different sizes. Simulations of the two applications are performed with the same solver in the OpenFOAM® computational toolbox for fluid dynamics.

The mixing indices are computed from a grid of sampling volumes. The choice of the size of the sampling volume depends on the relevant length scales of the system. There are three characteristic length scales: (a) The physical length scale, which corresponds to the solid volume fraction field, (b) the computational length scale that corresponds to the mesh size on which the CFD simulations are performed, and (c) the length scale corresponding to the sampling volume. The mixing indices can be applied directly to the same computational mesh used for CFD, which implies that the sampling volume length scale is equal to the computational length scale. However, larger sampling volumes could be used. For sampling volumes larger than the computational length scale but smaller than the smallest physical length scale, we expect the local mixing indices to be independent of sampling volume size.

Mixing can vary depending on the scale at which the system is observed and can be characterized by a sampling volume. If the sampling volumes larger than the smallest physical length scales are chosen, micro-mixing features may be excluded, and mixing at larger scales can be studied. The new mixing indices account for the phenomena of scale-dependent mixing. The performance of scale-dependence of the mixing index is characterized using synthetic fields as the data obtained from multi-fluid CFD simulations.

The conventional mixing indices such as Lacey and Rose are also computed from the sampling volume grid. They provide a single scalar measure of mixing in a system and can be called global mixing indices. We demonstrate the use of these indices with solid volume fraction and the solid fractions for gas-solid and solid-solid mixing, respectively. We extend these variance-based indices to be used with the local mixing index fields and find considerable improvement when using the global Rose solid mixing index, as opposed to the Rose solid fraction index.

A new performance criterion called the sensitivity of a mixing index is introduced. The local mixing indices developed have tunable sensitivity. Depending on the application, only a portion of the range of solid volume fraction or solid fraction can be accessed. By choosing an adequate accessed region, the mixing index can capture more details of local relative mixing and segregation. Consequently, the sensitivity of global indices based on the standard deviation or variance of the local solid mixing index is influenced. The sensitivity of the global indices can also be influenced by the choice of reference distributions corresponding to the perfectly mixed and perfectly unmixed states.

References

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