Several macroscale models with varying levels of complexity and covering macroscale phenomena such as the reactor residence time effects and micro- and macromixing behavior have been developed for olefin polymerization but is not discussed here in detail for the sake of brevity [149-157], [Pg.101]

Figure 8. The macroscale model of through-diffusion experiment. |

The transport equations appearing in macroscale models can be derived from the kinetic equation using the definition of the moment of interest. For example, if the moment of interest is the disperse-phase volume fraction, then it suffices to integrate over the mesoscale variables. (See Section 4.3 for a detailed discussion of this process.) Using the velocity-distribution function from Section 1.2.2 as an example, this process yields [Pg.21]

Figure 1.5. From the kinetic equation to macroscale models using moment methods that are based on reconstruction of the NDF. |

During the last decade considerable attention has been put on the macroscale modeling of bubble breakage in gas-liquid dispersions (e.g., [92, 43, 99, [Pg.825]

This example illustrates the classical problem faced when working with macroscale models (Struchtrup, 2005). No matter how the transport equations for the moments are derived, they will always contain unclosed terms that depend on higher-order moments (e.g. Up depends on 0p, etc.). In comparison, the solution to the kinetic equation for the NDF contains information about all possible moments. In other words, if we could compute n t, x, v) directly, it would not be necessary to work with the macroscale model equations. The obvious question then arises Why don t we simply solve the kinetic equation for the mesoscale model instead of working with the macroscale model [Pg.22]

M.F. Horstemeyer et al Using a micromechanical finite element parametric study to motivate a phenomenological macroscale model for void/crack nucleation in aluminum with a hard second phase. Mech. Matls. 35, 675-687 (2003) [Pg.131]

The importance of developing appropriate quantitative descriptions of microstructure that not only can be obtained by well-established stereological measurements but also can be expressed in a manner compatible with continuum descriptions of matter is fundamental to connecting the micro- and mesoscopic to the macroscopic descriptions of materials. Furthermore, such descriptions must be incorporated into macroscale models of material behavior in a manner that permits the calculation of macroscopic engineering properties in terms of the mesoscopic attributes of materials. [Pg.28]

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