Fluid model equations

Andrews, A. T. IV., and Sundaresan, S. Closures for filtered two-fluid model equations of gas-particle flows, Manuscript in preparation (2006). 

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

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... 

In reaction engineering the 2D and 3D multi-fluid model equations used are obviously solvable although with certain struggle, hence to some extent affected by the mathematical model properties reflected by the simplified model equations examined in the above mentioned work. Besides, in some cases numerical problems occur due to the inadequate mathematical properties reflected by the constitutive equations used in particular in the limit as —> 0... 

Lathouwers D, van den Akker HEA (1996) A numerical method for the solution of two-fluid model equations. In Numerical methods for multiphase flows. FED-Vol 236, ASME 1996 Fluids Engineering Division Conference, Volume 1, ASME 1996, pp 121-126... 

In this chapter several numerical methods frequently employed in reactor engineering are introduced. To simulate the important phenomena determining single- and multiphase reactive flows, mathematical equations with different characteristics have to be solved. The relevant equations considered are the governing equations of single phase fluid mechanics, the multi-fluid model equations for multiphase flows, and the population balance equation. 

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

The basic discretization of the two-fluid model equations is similar to the approximations of the corresponding transport equations for single phase flow. A minor difference is that the two-fluid model equations contain the novel phase fraction variables that have to be approximated in an appropriate manner. More important, to design robust, stable and accurate solution procedures with appropriate convergence properties for the two-fluid model equations, emphasis must be placed on the treatment of the interface transfer terms in the phasic momentum, heat and mass transport equations. Because of these extra terms, the coupling between the different equations is even more severe for multiphase flows than for single phase flows. 

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. 

No significant differences in convergence performance are generally observed assessing the above mentioned alternative schemes for the solution of the two-fluid model equations [80]. A main limitation associated with these 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... 

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]. 

Bove [16] proposed a different approach to solve the multi-fluid model equations in the in-house code FLOTRACS. To solve the unsteady multifluid model together with a population balance equation for the dispersed phases size distribution, a time splitting strategy was adopted for the population balance equation. The transport operator (convection) of the equation was solved separately from the source terms in the inner iteration loop. In this way the convection operator which coincides with the continuity equation can be employed constructing the pressure-correction equation. The population balance source terms were solved In a separate step as part of the outer iteration loop. The complete population balance equation solution provides the... 

The shear-dependent viscosity of a commercial grade of polypropylene at 403 K can satisfactorily be described using the three constant Ellis fluid model (equation 1.15), with the values of the constants fiQ = 1.25 x lO Ea s, Ti/2 = 6900 Pa and a = 2.80. Estimate the pressure drop required to maintain a volumetric flow rate of 4cm /s through a 50 mm diameter and 20 m long pipe. Assume the flow to be laminar. 

Rovinsky et al. compared the solutions for laminar-laminar stratified flows obtained via the exact model with those predicted by the two-fluid model. Equations 11-14 [12] It was found that the error in H/D is bounded by 2.5%, and a typical error in the system pressure drop is about 10%. The comparison shows that for laminar-laminar flows, in the range of 0.1 < < 10 and 10" < xyiJ < 10, the accuracy... 

Starting out from the governing two-fluid model equations with the KTGF closure, as deflned earlier by (4.228)-(4.248), several model simplifications were imposed. First, the particle phase collisional pressure gradient was approximated by a semi-empirical elasticity modulus parameterization on the form ... 

Chao et al. [22] did simulate dynamic size segregation in bubbling fluidized bed systems by use of exactly the same three-fluid model equations (4.413)-(4.408) as employed by Chao et al. [23]. Introductory, the empirical friction particle-particle drag coefficient relation proposed Chao et al. [19] was employed. 

The average multi-fluid model equations are outlined in the following together with the conventional interfacial closures that are frequently adopted in gas-liquid bubbly flow analyzes. The average multi-fluid continuity equation for phase k reads ... 

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