Hyperbolic Homogenized Models for Thermal and Solutal Dispersion
Hsueh‐Chia Chang, Vemuri Balakotaiah · SIAM Journal on Applied Mathematics · 2003
We formulate a general theory, based on a Lyapunov--Schmidt expansion, for averaging thermal and solutal dispersion phenomena in multiphase reactors, with specific attention to the important Taylor mechanism due to transverse intraphase and interphase capacitance-weighted velocity gradients. We show that the classical Taylor dispersion phenomena are better described in terms of low dimensional models that are hyperbolic and contain an effective local time or length scale in place of the traditional Taylor dispersion coefficient. This description eliminates the use of an artificial exit boundary condition associated with parabolic homogenized equations as well as the classical upstream-feedback and infinite propagation speed anomalies. Our approach is also applicable for describing steady dispersion in the presence of reaction and thermal generation or consumption. For two-phase systems, maximum dispersion is found to exist at an optimum fraction $\epsilon _{f}$ of the lower-capacitance phase. For the disparate phase capacities of most reactors, thermal or solutal dispersion is shown to have the scaling $\frac{\epsilon _{f}p^{2}}{(1-\epsilon _{f})\Gamma}\alpha _{f}$, where $\alpha_{f}$ is the thermal diffusivity of the low-capacitance phase, $\Gamma$ is the capacitance ratio, and p is the transverse Peclet number.