Hyperbolic equation moments

Appendix C. Moment methods with hyperbolic equations... 

This result was first derived by Aris (1956) using the method of moments. While the resulting model now includes both the effects (axial molecular diffusion and dispersion caused by transerverse velocity gradients and molecular diffusion) it has the same deficiency as the Taylor model, i.e. converting a hyperbolic model into a parabolic equation. 

In summary, although the weakly hyperbolic nature of Eq. (8.7) has been shown rigorously only for ID phase space (i.e. one velocity component in the KE), experience strongly suggests that the full 3D system is also weakly hyperbolic. This observation implies that the numerical schemes used to solve the moment-transport equations closed with QBMM must be able to handle local delta shocks in the moments. Qne such class of numerical schemes consists of the kinetics-based finite-volume solvers presented in Section 8.2. As a final note, we should mention that the work of Chalons et al (2012) using extended Gaussian quadrature (see Section 3.3.2) and kinetics-based finite-volume solvers to close Eq. (8.7) suggests that the system with 2A + 1 moments is fully hyperbolic and thus does not exhibit... 

The application of QBMM to Eq. (C.l) will require a closure when m(7 depends on 7 Nevertheless, the resulting moment equations (used for the QMOM or the EQMOM) and transport equations for the weights and abscissas (used for the DQMOM) will still be hyperbolic. In terms of hyperbolic conservation laws, the moments are conserved variables (which result from a linear operation on /), while the weights and abscissas are primitive variables. Because conservation of moments is important to the stability of the moment-inversion algorithms, it is imperative that the numerical algorithm guarantee conservation. For hyperbolic systems, this is most easily accomplished using finite-volume methods (FVM) (or, more specifically, realizable FVM). The other important consideration is the accuracy of the moment closure used to close the function, as will be described below. 

The zero, first and second moment of the transient system can be calculated via the Laplace transform [7,8]. With a Thiele modulus greater than five, e.g. strong difhision limitation, the hyperbolic functions in the moments equations tend to their asymptotic values. It can be shown theoretically that the moments become linearly dependent and the number of model parameters reduces to two, where DrHr = constant and kr. If additionally the effect of adsorption on the dynamic response becomes negligible, the number of independent parameters, which are needed to describe the response curve, reduces to one DrH,kr = constant. The latter case can therefore not extract more parameters than could be obtain from a steady state measurement. 

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