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.
Dispersion of a non-uniform slug in time-dependent flow
Dispersion of a non-uniform slug in time variable fully developed laminar flow is studied by an exact method which in principle is valid for all values of time. A generalized dispersion model, which involves an infinite set of time-dependent coefficients Ki(T), arises naturally as a consequence of the analysis. Expressions are obtained from first principles for the convective coefficient K1(T), the apparent diffusion coefficient K2(T), and the rest of the coefficients Ki(T). The convective coefficient K1(T) is a new quantity which is time dependent, even if the velocity field is steady, because the initial distribution of solute is non-uniform. The general method of analysis proposed is applied to the particular case of a slug of material of length xs and radius aR which is initially symmetrically located in a tube of radius R, and then is dispersed in steady laminar flow. Breakthrough curves for various downstream positions show that the mean velocity of the material in the slug is greater than the average velocity of flow for small residence times and, as one would expect intuitively, approaches this value as the distance between the injection and observation points increases. It also is shown that for T ≲ 0.5, K2, the axial dispersion coefficient, decreases as the initial slug radius decreases. However, for all values of 'a', both K1 and K2 approach the asymptotic values of —½ and (Pe)-2 + 1/192 respectively for values of dimensionless time T ≳ 0.5. Since the initial concentration distribution of solute in the slug affects the time of arrival of the peak mean concentration at a downstream location, this may cause errors in the estimation of mean flow rates by tracer techniques. For example, if one injects a quantity of material into a stream and assumes it is uniformly distributed across the tube, when in reality it occupies only a part of the tube, this leads to overestimating the average flow velocity. This may be of practical importance in a variety of applications.
An asymptotic theory for dispersion of reactive contaminants in parallel flow
Abstract The classic problem, first treated by Taylor [18], of the dispersion of inert soluble matter in fluid flow continues to attract attention from researchers describing the approach to the asymptotic state [5, 17]. The present article considers some of the complications caused when the solute is chemically active, Dispersing chemically active solutes occur in diverse fields such as chromatography, chemical engineering and environmental fluid mechanics. The asymptotic large-time analysis of Chatwin [5] is re-worked to handle the case of reactive solutes dispersing in parallel flow. Matching between moderate and large-time solutions requires consideration of the integral moments of the reactive contaminant could, and the Aris method of moments is therefore invoked and modified for reaction effects. The results are applied in detail to the outstanding paractical example—the chemical flow reactor (a device used to measure reaction rates for chemical reactions taking place between fluids). For this case, the paper provides a practical alternative to recent variable diffusion coefficient studies [6, 7, 15], and presents further results for the concentration distribution and for the limiting behaviour under weak and vigorous recactions at the boundary of the flow.