Dilute-phase momentum balance

Dilute-phase momentum balance all effective particle weight in unit dilute-phase volume is balanced by the fluid drag ... 

Dense-phase momentum balance effective particle weight in unit dense-phase volume is partially supported by the dense-phase fluid flow, and the rest is supported by the bypassing dilute phase fluid flow,... 

Three types of theoretical approaches can be used for modeling the gas-particles flows in the pneumatic dryers, namely Two-Fluid Theory [1], Eulerian-Granular [2] and the Discrete Element Method [3]. Traditionally the Two-Fluid Theory was used to model dilute phase flow. In this theory, the solid phase is being considering as a pseudo-fluid. It is assumed that both phases are occupying every point of the computational domain with its own volume fraction. Thus, macroscopic balance equations of mass, momentum and energy for both the gas and the solid... 

Tomiyama [148] and Tomiyama and Shimada [150] adopted a N + 1)-fluid model for the prediction of 3D unsteady turbulent bubbly flows with non-uniform bubble sizes. Among the N + l)-fluids, one fluid corresponds to the liquid phase and the N fluids to gas bubbles. To demonstrate the potential of the proposed method, unsteady bubble plumes in a water filled vessel were simulated using both (3 + l)-fluid and two-fluid models. The gas bubbles were classified and fixed in three groups only, thus a (3 + 1)- or four-fluid model was used. The dispersions investigated were very dilute thus the bubble coalescence and breakage phenomena were neglected, whereas the inertia terms were retained in the 3 bubble phase momentum equations. No population balance model was then needed, and the phase continuity equations were solved for all phases. It was confirmed that the (3 + l)-fluid model gave better predictions than the two-fluid model for bubble plumes with non-uniform bubble... 

The pseudo-homogeneous or dusty-gas model very small particle Stokes number and limited polydispersity (momentum-balance equation only for the continuous phase if the system is dilute or for the mixture of continuous and disperse phases if the system is dense). 

This way, the intuitively anticipated diffusion of bubbles from a region with many bubbles to a dilute region is back into the theory not in the mass balances, but rather in the momentum balances, as it is after all a consequence of the drag force. The diffusion coefficient in Eq. (3.11) is estimated from the ratio of turbulence and bubble response times and the turbulent kinetic energy of the liquid phase. Finally, for the drag coefficient various relations are present, for example, based on Schiller and Naumann [47] ... 

Theoretical representation of the behaviour of a hydrocyclone requires adequate analysis of three distinct physical phenomenon taking place in these devices, viz. the understanding of fluid flow, its interactions with the dispersed solid phase and the quantification of shear induced attrition of crystals. Simplified analytical solutions to conservation of mass and momentum equations derived from the Navier-Stokes equation can be used to quantify fluid flow in the hydrocyclone. For dilute slurries, once bulk flow has been quantified in terms of spatial components of velocity, crystal motion can then be traced by balancing forces on the crystals themselves to map out their trajectories. The trajectories for different sizes can then be used to develop a separation efficiency curve, which quantifies performance of the vessel (Bloor and Ingham, 1987). In principle, population balances can be included for crystal attrition in the above description for developing a thorough mathematical model. 

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