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This work presents a brief study about mathematical modeling of two-phase flow (liquid + gas), approaching homogeneous phase flows and heterogeneous phase flows, in addition to mathematical modeling of pressure drop in flow restriction through abrupt expansions and abrupt contractions. Also presents a summary of flow patterns main types and a brief study about how the flow velocity influences these patterns.

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In this paper it is addressed the prediction of form and location of detached shock waves ahead of two-dimensional and axially symmetric bodies at an zero angle of attack. As shown in Figs. 5 and 6, results show a very good agreement with experimental data. In this context an approximate method, based on a simplified form of the continuity relation, is developed to predict the location of detached shock waves ahead of two-dimensional and axially symmetric bodies. In order to reduce the problem to an equivalent onedimensional form, it is assumed that: (1) The form of the shock between its foremost point and its sonic point is adequately represented by a hyperbola asymptotic to tile free-stream Mach lines; and (2) the sonic line between the shock and the body is straight and inclined at a constant angle. Although the new methodology has some points of contact with earlier methodologies, the novelty here is that it is used Missile Datcom code as an aid to find out sonic point on body and also it is adopted Parametric System Identification (PSI) in the determination of bow shock shape which uses the Matlab® optimizer fmincon function and an active set strategy to minimize an error in a rms sense subject to simple constraint placed on the parameters by the user. The optimizer function calls a user written function which calculates the shape of the shock wave using the current parameters supplied by optimizer. Also for the shock distance L, the methodology presented here allows to select the value of the mentioned constant angle consistently based either in aerodynamics literature or through physical considerations. As the L value is previously known from measurements or aerodynamics literature, it was used an optimizer to minimize the error between predicted and known result varying a parameter which absorbs all inconsistencies that arise when it is used the basic Moeckel’s model considered here. Once the principal characteristics of the shock wave are calculated, an error value is returned to optimizer function based on the differences between predicted and known results.

In this paper, a model based on the Eulerian-Eulerian approach considering the kinetic theory of granular flow (KTGF) is used in order to provide a comprehensive comparison between Gidaspow (1994) drag model and recent models found in literature, such as Hill et al. (2001), Yang et al. (2003), Zhang-Reese (2003), Van der Hoff et al. (2005) and Beetstra et al. (2007). The effects of these drag models on hydrodynamics behavior of gas-solid flow in fluidized bed will be investigated by using the MFIX code. The results are used to assess their capacity in predicting parameters such as pressure drop, bed expansion and voidage profiles, and then finally validated with the experimental results of Taghipour et al. (2005), which is available in literature. Results show that Hill et al. (2001), Zhang-Reese (2003) and Beetstra et al. (2007) drag models can be used to predict hydrodynamic parameters of gas-solid flow in a fluidized-bed much more accurately than Gidaspow (1994) model.