Formulation with Discrete Sources

For a two-layered particle as shown in Fig. 2.2, the null-field equations formulated in terms of distributed vector spherical wave functions (compare to (2.70), (2.71) and (2.72)) 

For a multilayered particle, it is apparent that the solution methods with distributed sources use essentially the same matrix equations as the solution methods with locafized sources. The matrices A, A and A are given by (2.109), (2.110) and (2.112), respectively, with Qf in place of Qf, while 

The expressions of the elements of the Q matrix are given by (2.84)-(2.87) with the localized vector spherical wave functions replaced by the distributed vector spherical wave functions. The transition matrix is 

The use of distributed vector spherical wave functions improves the numerical stability of the null-field method for highly elongated and flattened layered particles. Although the above formalism is valid for nonaxisymmetric particles, the method is most effective for axisymmetric particles, in which case the 2 -axis of the particle coordinate system is the axis of rotation. Applications of the null-field method with distributed sources to axisymmetric layered spheroids with large aspect ratios have been given by Doicu and Wriedt [50]. 

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The radiative source term is a discretized formulation of the net radiant absorption for each volume zone which may be incorporated as a source term into numerical approximations for the generalized energy equation. As such, it permits formulation of energy balances on each zone that may include conductive and convective heat transfer. For K—> 0, GS —> 0, and GG —> 0 leading to S —> On. When K 0 and S = 0N, the gas is said to be in a state of radiative equilibrium. In the notation usually associated with the discrete ordinate (DO) and finite volume (FV) methods, see Modest (op. cit., Chap. 16), one would write S /V, = K[G - 4- g] = Here H. = G/4 is the average flux... 

Lee et al [66] and Prince and Blanch [92] adopted the basic ideas of Coulaloglou and Tavlarides [16] formulating the population balance source terms directly on the averaging scales performing analysis of bubble breakage and coalescence in turbulent gas-liquid dispersions. The source term closures were completely integrated parts of the discrete numerical scheme adopted. The number densities of the bubbles were thus defined as the number of bubbles per unit mixture volume and not as a probability density in accordance with the kinetic theory of gases. 

Comparing the source term expressions (9.76) to (9.79) with (9.2) to (9.5) it is clearly seen that only under particular conditions will the two formulations give rise to identical expressions for the source terms. The macroscopic formulation is explicitly expressed in terms of a discrete discretization scheme and is very difficult to convert to other schemes. 

Only a few continuous source term closures are available, hence the discrete PBE model closures are used in practice. The macroscopic statistical mechanical PBE model thus coincides with the macroscopic PBE derived from continuum mechanical principles. In this way there are little or no differences in employing these two approaches. However, the formulation of the constitutive equations are strongly influenced by the concepts of kinetic theory of dilute gases. Nevertheless, the present closures are still at an early stage of development and future work should continue developing more reliable pa-rameterizations of the kernels. These must be validated for the application in question. 

For bubbly flows most of the early papers either adopted a macroscopic population balance approach with an inherent discrete discretization scheme as described earlier, or rather semi-empirical transport equations for the contact area and/or the particle diameter. Actually, very few consistent source term closures exist for the microscopic population balance formulation. The existing models are usually solved using discrete semi-integral techniques, as will be outlined in the next sub-section. 

In the Lagrangian approach, the elemental control volume is considered to be moving with the fluid as a whole. In the Eulerian approach, in contrast, the control volume is assumed fixed in the space, the fluid is assumed to flow through and pass the control volume. The particle-phase equations are formulated in Lagrangian form, and the coupling between the two phases is introduced through particle sources in the Eulerian gas-phase equations. The standard k-e turbulence model, finite rate chemistry, and DTRM (discrete transfer radiation model) radiation model are used. 

The projection-based model order reduction algorithm begins with a spatial discretization of the governing PDEs to attain the dynamic system equations as Eq. 11. Specifically, here, X(t) is the state vector of unknowns (a function of time) on the discrete nodes, n is the total number of nodes A is formulated by the numerical discretization Z defines the functions of boundary conditions and source terms and B relates the input function to each state X. Equation 11 can be recast into the frequency domain in terms of transfer function T(s). T(s) then is expanded as a Taylor series at s = 0 yielding... 

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