Algebraic-slip mixture model

In this section we derive the algebraic-slip mixture model equations for cold flow studies starting out from the multi-fluid model equations derived applying the time- after volume averaging operator without mass-weighting [204, 205]. The momentum equations for the dispersed phases are determined in terms... 

For the relative (slip) velocity numerous empirical correlations are available in the literature [34]. Hence it follows that each component of the dispersed phase momentum equation is reduced to an algebraic-slip relation (3.428). This is the reason why this mixture model formulation is referred to as the algebraic-slip mixture model. 

When the diffusion velocity is eliminated, the main difference between the particular diffusion- and algebraic-slip mixture models presented in this... 

The main contribution from the work of Luo [95, 96] was a closure model for binary breakage of fluid particles in fully developed turbulence flows based on isotropic turbulence - and probability theories. The author(s) also claimed that this model contains no adjustable parameters, a better phrase may be no additional adjustable parameters as both the isotropic turbulence - and the probability theories involved contain adjustable parameters and distribution functions. Hagesaether et al [49, 50, 51, 52] continued the population balance model development of Luo within the framework of an idealized plug flow model, whereas Bertola et al [13] combined the extended population balance module with a 2D algebraic slip mixture model for the flow pattern. Bertola et al [13] studied the effect of the bubble size distribution on the flow fields in bubble columns. An extended k-e model was used describing turbulence of the mixture flow. Two sets of simulations were performed, i.e., both with and without the population balance involved. Four different superficial gas velocities, i.e., 2,4,6 and 8 (cm/s) were used, and the superficial liquid velocity was set to 1 (cm/s) in all the cases. The population balance contained six prescribed bubble classes with diameters set to = 0.0038 (m), d = 0.0048 (m), di = 0.0060 (m), di = 0.0076 (m), di = 0.0095 (m) and di = 0.0120 (m). 

Chen et al [12] and Bertola et al [8] simulated mixtures consisting of A1+1 phases by use of algebraic slip mixture models (ASMMs) which have been combined with a population balance equation. Each bubble size group did have individual local velocities which were calculated from appropriate algebraic slip velocity parameterizations. In order to close the system of equations, the mixture velocity was expressed in terms of the individual phase velocities. The average gas phase velocity was then determined from a volume weighted slip velocity superposed on the continuous phase velocity. Chen et al [12] also did run a few simulations with the ASMM model with the same velocity for all the bubble phases. 

ADI Alternating Direction Implicit ADM Axial Dispersion Model AS MM Algebraic Slip Mixture Model... 

The most popular simplified version of n-fluid model is the mixture model. In the mixture model, different phases share the same temperature and pressure profiles. However, the velocity field is different for each phase. In this model, conservation equations are solved for the mixture, and the velocity difference between phases is handled through the concept of slip velocity, which is the difference between the velocities of two phases. In the commercially available software FLUENT, this slip velocity is found through the algebraic formulation proposed by Manninen et al. (1996). The basic assumption of the algebraic slip mixture model is that a local equilibrium between the phases should be reached over short spatial length scale ... 

Gas sparging can be modeled using the Eulerian multiphase model or the algebraic slip mixture model. For the Eulerian multiphase model, two sets of momentum equations are used, and the same comments regarding the slip velocity between phases apply, although the issue is not as critical. That is, the velocity data used for the gas phase could be corrected slightly from the liquid-phase velocities but need not be because the gas phase has so little inertia compared to the liquid phase. When the algebraic slip mixture model is used, separate boundary conditions are not required for the individual phases, so a correction of the velocity data is not required. 

Alternating Direction Implicit Axial Dispersion Model Algebraic Slip Mixture Model Name of Commercial CFD Code Bubbling Bed... 

Algebraic-Slip-Mixture model to model the interactions between water, air and sludge... 

The mixture model is based on the solution of a single mixture momentum equation for all phases, which significantly reduces computational effort. Saalbach and Hunze (2008) used the Algebraic-Slip-Mixture model to simulate the interactions of three phases water, air and sludge in pilot and full-scale plants. The mixture model could account for the slip velocities of the dispersed phase and the continuous phase relative to the mixture. 

To determine the dispersed phase velocities as occurring in the phasic continuity equations in both formulations, the momentum equation of the dispersed phases are usually approximated by algebraic equations. Depending on the concept used to relate the phase k velocity to the mixture velocity the extended mixture model formulations are referred to as the algebraic slip-, diffusion- or drift flux models. 

In the Homogeneous Algebraic Slip model [4] applied to gas-liquid systems, governing equations for mixture quantities are solved for rather than for phase-specific quantities u = Y UkPkUk /atPk,p = Pm, Ud = Uo-u, This implies... 

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