Wave model for longitudinal dispersion: Application to the laminar‐flow tubular reactor
The wave model for longitudinal dispersion, published elsewhere as an alternative to the commonly used dispersed plug-flow model, is applied to the classic case of the laminar-flow tubular reactor. The results are compared in a wide range of situations to predictions by the dispersed plug-flow model as well as to exact numerical calculations with the 2-D model of the reactor and to other available methods. In many practical cases, the solutions of the wave model agree closely with the exact data. The wave model has a much wider region of validity than the dispersed plug-flow model, has a distinct physical background, and is easier to use for reactor calculations. This provides additional support to the theory developed elsewhere. The properties and the applicability of the wave model to situations with rapidly changing concentration fields are discussed. Constraints to be satisfied are established to use the new theory with confidence for arbitrary initial and boundary conditions.
Equivalence of bicontinuum and second‐order transport in heterogeneous soils and aquifers
The mathematical equivalence of a closed second‐order transport equation for reactive solutes in saturated heterogeneous porous media (e.g., soils and aquifers) and a two‐region, mobile‐mobile, formulation is demonstrated in two ways: (1) by averaging the bicontinuum equations and (2) by transforming the second‐order equation to canonical form. The derivation of this equivalence is limited to media with heterogeneities in a plane orthogonal to the flow direction which represents a bidirectional generalization of the stratified case. Results indicate how the parameters of the bicontinuum formulation can be obtained from statistical descriptions of this heterogeneity and establish a stochastic interpretation for the bicontinuum equations. One‐dimensional analytical solutions of the equivalent transport equations are compared to a three‐dimensional numerical simulation of transport in a spatially autocorrelated lognormal random conductivity field. Results indicate that the upscaled equations provide good approximations of the dynamics of the mean resident concentration, the flux‐averaged concentration, and their variances.
The early stages of contaminant dispersion in shear flows
The dispersion of a spot of contaminant in a high-Péclet-number laminar flow is studied by means of the ray method developed by Cohen & Lewis (1967). This method is free from the usual severe restrictions on the time range. Thus, it is possible to investigate strong shear-distortions of the concentration distribution. Also, the effects of rigid boundaries can be allowed for simply by including reflected rays. Three examples are studied in detail: stagnation flow, a point vortex and plane Poiseuille flow.