Solving the Two-Fluid Model Equations

No significant differences in convergence performance are generally observed assessing the alternative SIMPLE/SIMPLEC/SIMPLER, volume fraction and pressure correction discretization schemes for the solution of the two-fluid model equations [94]. A main limitation associated with these volume fraction and pressure correction procedures is that it is impossible to ensure that both the overall mixture continuity and the individual phase continuities are satisfied at the same time. In addition, to solve the two-fluid model problems, an implicit treatment of the interphase coupling terms is required. In the modern solution algorithms for the two-fluid 

The velocity- and pressure correction equations in IPSA are frequently derived using the SIMPLEC method (i.e., the SIMPLE- Consistent approximation) by van Doormal and Raithby [239]. 

Detailed and illustrative descriptions of the commonly used two-fluid model solution algorithms and discretization schemes employed in order to solve dynamic two-fluid models for reactive systems can be found elsewhere [94, 137, 218]. 

In multiphase reactive flows, the interfacial transfer fluxes of momentum, heat and species mass are of great importance. These interfacial transfer fluxes are generally modeled as a product of the interfacial area concentration, the driving force denoting the difference in the phase values of the primitive variables, and the transfer (proportionality) coefficients. Mathematically, a generic flux 4 can be expressed on the form  

In three-dimensional problems, the neighbor coefficients are arranged in two subdiagonals located next to the main diagonal and four peripherical diagonals  

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Explicit Fractional Step Algorithm for Solving the Two-Fluid Model Equations Applied to... 

Special Challenges in Solving the Two-fluid Model Equations... 

A property of the finite volume method is that numerous schemes and procedures can be designed in order to solve the two-fluid model equations. In addition, the coupling terms can be approximated and manipulated in different ways. Besides, it is very difficult to predict the convergence and stability properties of novel solution methods. These aspects collectively increase the possibility of devising a numerical procedure which, when put to the test, does not converge. 

In the modern solution algorithms for the two-fluid model equations, the latter problem is generally relaxed by coupled strategies solving the corresponding transport equations of both phases simultaneously [90, 103, 104, 99, 98, 6, 119]. 

These models require information about mean velocity and the turbulence field within the stirred vessels. Computational flow models can be developed to provide such fluid dynamic information required by the reactor models. Although in principle, it is possible to solve the population balance model equations within the CFM framework, a simplified compartment-mixing model may be adequate to simulate an industrial reactor. In this approach, a CFD model is developed to establish the relationship between reactor hardware and the resulting fluid dynamics. This information is used by a relatively simple, compartment-mixing model coupled with a population balance model (Vivaldo-Lima et al., 1998). The approach is shown schematically in Fig. 9.2. Detailed polymerization kinetics can be included. Vivaldo-Lima et a/. (1998) have successfully used such an approach to predict particle size distribution (PSD) of the product polymer. Their two-compartment model was able to capture the bi-modal behavior observed in the experimental PSD data. After adequate validation, such a computational model can be used to optimize reactor configuration and operation to enhance reactor performance. 

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... 

Several extensions of the two-fluid model have been developed and reported in the literature. Generally, the two-fluid model solve the continuity and momentum equations for the continuous liquid phase and one single dispersed gas phase. In order to describe the local size distribution of the bubbles, the population balance equations for the different size groups are solved. The coalescence and breakage processes are frequently modeled in accordance with the work of Luo and Svendsen [74] and Prince and Blanch [92]. 

In order to close the two-fluid model, constitutive equations are required for (i) stresses (Table 4.2), (ii) internal heat transfer (Table 4.3), (iii) internal mass transfer (Table 4.4), (iv) interfacial heat transfer (Table 4.5), (v) interfacial momentum transfer (Table 4.6), and (vi) solid phase collision pressure (Table 4.7). To solve the mathematical model, the finite volume discretization technique was employed. 

For other discussions of two-phase models and numerical solutions, the reader is referred to the following references thermofluid dynamic theory of two-phase flow (Ishii, 1975) formulation of the one-dimensional, six-equation, two-phase flow models (Le Coq et al., 1978) lumped-parameter modeling of one-dimensional, two-phase flow (Wulff, 1978) two-fluid models for two-phase flow and their numerical solutions (Agee et al., 1978) and numerical methods for solving two-phase flow equations (Latrobe, 1978 Agee, 1978 Patanakar, 1980). 

The number of equations to be solved is, among other things, related to the turbulence model chosen (in comparison with the k-e model, the RSM involves five more differential equations). The number of equations further depends on the character of the simulation whether it is 3-D, 21/2-D, or just 2-D (see below, under The domain and the grid ). In the case of two-phase flow simulations, the use of two-fluid models implies doubling the number of NS equations required for single-phase flow. All this may urge the development of more efficient solution algorithms. Recent developments in computer hardware (faster processors, parallel platforms) make this possible indeed. 

Computational fluid dynamics were used to describe the flow which undergoes a fast transition from laminar (at the fluid outlets) to turbulent (in the large mixing chamber) [41]. Using the commercial tool FLUENT, the following different turbulence models were applied a ke model, an RNC-ki model and a Reynolds-stress model. For the last model, each stream is solved by a separate equation for the two first models, two-equation models are applied. To have the simulations at... 

The main approach for modelling multiphase flows has been through solving conservation equations described in terms of Eulerian phase-averaged mean quantities - the two-fluid approach [665], The Eulerian mean velocity in a control volume V (such as the volume within the perimeter S of the cloud of particles) is defined as the velocity ux (for each component) averaged over the volume occupied by the fluid (ie the fluid space between the bodies),... 

Rocha and Paixao [38] proposed a pseudo two-dimensional mathematical model for a vertical pneumatic dryer. Their model was based on the two-fluid approach. Axial and radial profiles were considered for gas and solid velocity, water content, porosity, temperatures, and pressure. The balance equations were solved numerically using a finite difference method, and the distributions of the flow field characteristics were presented. This model was not validated with experimental results. 

Above-mentioned reaction, diffusion and advection influence mass transfer in rock-water system. It is generally difficult to solve the differential equation including all these mechanisms. Thus, the two coupled models at constant temperature and pressure will be explained below. They are (1) reaction-fluid flow model, (2) reaction-diffusion model, (3) diffusion-fluid flow model. In addition to these coupled models, model taking into account the change in temperature will be considered. 

For each fluid we could write an equation such as Eqn. (14.10). If the heat capacity of the wall would be ignored, we would get a set of two simultaneous equations that could be solved. The outlet temperature responses of the two fluids would consist of a summation of exponential functions, which are an indication of higher order models. An approximate solution of the set of two differential equations is discussed, among others, by Friedly (1972), Harriott (1%4) andMozley (1956). 

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