We consider a heap of grains driven by gravity down an incline. We assume that the heap is supported at its base on a relatively thin carpet of intensely sheared, highly agitated grains that interact through collisions. We adopt the balance laws, constitutive relations, and boundary conditions of a kinetic theory for dense granular flows and determine the relationship between the shear stress, normal stress, and relative velocity of the boundaries in the shear layer in an analysis of a steady shearing flow between identical bumpy boundaries. This relationship permits us to close the hydraulic equations governing the evolution of the shape of the heap and the velocity distribution at its base. We integrate the resulting equations numerically for typical values of the parameters for glass spheres. (c) 1999 American Institute of Physics.

We consider the region of agitated grains that is at rest in the neighbourhood of its interface with a dense granular flow. We suppose that this region can be modelled as an amorphous solid of nearly elastic spheres in which both momentum and energy are transferred and energy is dissipated in collisions. Making rather rough assumptions about the collision probability, we calculate the stress and energy flux in the solid and use their continuity at the interface to obtain boundary conditions on the flow. We employ them with existing kinetic theory for nearly elastic spheres to solve boundary-value problems for shearing between two such interfaces and between such an interface and a flat plate to which spheres have been rigidly attached. For the latter, we compare the predictions of the theory with the results of experiments.

In the present paper, two gas‐liquid stirred tanks, one agitated by a radial impeller and another by an axial impeller, are modelled using the open‐source computational fluid dynamic (CFD) package OpenFOAM (open source field operation and manipulation). The combined effect of the bubble break‐up and coalescence in the tank is considered by a population balance model (PBM) called extended quadrature method of moments (EQMOM). The three‐dimensional simulation is made using a multiple reference frame (MRF), a well‐established method for the modelling of mixers. Dispersed gas and bubble dynamics in the turbulent flow are modelled using the Eulerian‐Eulerian approach (E‐E) with mixture k‐epsilon turbulent model and the modified Tomiyama drag coefficient for the momentum exchange. The model is developed to predict the spatial distribution of gas phase fraction, Sauter mean bubble diameter (d32), number density function (NDF), dissolved oxygen (DO) evolution, and flow structure. The numerical results are compared with experimental data and a fair agreement is achieved. The results of the axial impeller are discussed based on four impeller rotational speeds with different volumetric mass transfer coefficients.

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Abstract:In order to optimize and design new bubbly flow reactors, it is necessary to predict the bubble behavior and properties with respect to the time and location. In gas-liquid flows, it is easily observed that the bubble sizes may vary widely. The bubble size distribution is relatively sharply defined, and bubble rises are uniform in homogeneous flow; however bubbles aggregate, and large bubbles are formed rapidly in heterogeneous flow. To assist in the analysis of these systems, the volume, size and other properties of dispersed bubbles can be described mathematically by distribution functions. Therefore, a mathematical modeling tool called the Population Balance Model (PBM) is required to predict the distribution functions of the bubble motion and the variation of their properties. In the present paper, two rectangular bubble columns and a water electrolysis reactor are modeled using the open-source Computational Fluid Dynamic (CFD) package OpenFOAM. Furthermore, the Method of Classes (CM) and Quadrature-based Moments Method (QBMM) are described, implemented and compared using the developed CFD-PBM solver. These PBM tools are applied in two bubbly flow cases: bubble columns (using a Eulerian-Eulerian two-phase approach to predict the flow) and a water electrolysis reactor (using a single-phase approach to predict the flow). The numerical results are compared with measured data available in the scientific literature. It is observed that the Extended Quadrature Method of Moments (EQMOM) leads to a slight improvement in the prediction of experimental measurements and provides a continuous reconstruction of the Number Density Function (NDF), which is helpful in the modeling of gas evolution electrodes in the water electrolysis reactor.