Jorge Ramírez-Muñoz

HYDRODYNAMIC INTERACTION OF TWO SPHERICAL BUBBLES RISING IN-LINE: A SEMI-ANALYTICAL APPROACH

A model is proposed to relate the flow structure in the wake behind a leading bubble with the hydrodynamic force acting on the trailing bubble rising in-line. The model is valid at separation distances between bubbles (s) within the same order of magnitude as their diameters (dB) (2.5 ≤ s/dB ≤ 14.5) and moderate-to-large particle Reynolds numbers (50 ≤ Re1 ≤ 300). An equation for the axial velocity profile in the wake of a spherical bubble was proposed in the form of an analytical approximation, but incorporating a theoretically reasoned artificial origin, which was fitted to numerical data. This equation, once substituted in a general hydrodynamic force model, resulted in a predictive equation explaining quantitatively the in-line interaction of a pair of bubbles in terms of the average axial velocity in the leading bubble wake. Moreover, the force experienced by the leading bubble due to the trailing bubble presence was obtained, and the theoretical equilibrium distances between the bubbles were calculated; good agreement with known behavior was achieved.

Modelo Hidrodinámico para la Velocidad de un Par de Burbujas Ascendiendo en Línea

Resumen Se presenta un analisis hidrodinamico de la interaccion de dos burbujas esfericas del mismo tamano alineadas en la direccion de un flujo uniforme, laminar, newtoniano e incompresible. La burbuja puntera se considera aislada y la burbuja acarreada (segunda burbuja) esta sujeta a la accion de cuatro fuerzas: flotacion, arrastre cuasiestacionario, impulso del fluido y masa agregada. Aplicando un balance de fuerzas sobre cada burbuja se obtuvo una nueva expresion para la velocidad de ascenso de la burbuja acarreada. La expresion se comparo con mediciones experimentales, realizadas por otros autores citados en la literatura, para numeros de Reynolds de 3.06, 21.5 y 35.4 y se obtuvo un error relativo promedio entre 1.2% y 2.1%. Se discuten dos alternativas para aproximar la reduccion del arrastre sobre la burbuja acarreada. Se encontro que el arrastre cuasiestacionario no puede incorporar el efecto total de la estela sobre la burbuja acarreada. Palabras clave: modelo hidrodinamico, velocidad de burbujas, e

Optimization of a Treatment System of Wastewater Streams for Electrochemical Cr(VI) Reduction: Selective versus Centralized Treatment

Abstract The problem of the optimization of selective treatment systems of wastewater streams contaminated with hexavalent chromium [Cr(VI)] is investigated. In order to comply with the Mexican environmental norm of Cr(VI) for treated wastewater streams at minimum cost, a nonlinear programming (NLP) model for the electrochemical reduction of hexavalent chromium to trivalent chromium was developed. The model incorporates a variable reaction rate, which is a function of the Cr(VI) concentration and the electrical current density of the electrochemical process. For this purpose, a basic superstructure of the effluent treatment is proposed. The superstructure is composed of three continuous electrochemical reactors without recirculation, and it may produce either a series and/or parallel design topology. The NLP model was used to minimize the objective function, defined as the total annual cost (TAC), which includes the capital cost of each electrochemical reactor, the electrical energy cost and the cost of the treatment of the wastewater streams. In order to investigate the solution set of the proposed NLP model, i. e., to improve the possibilities of obtaining optimum solutions based on economic criteria, a multi-start algorithm was implemented. Two example problems are used to show the versatility of the model and different local optimal solutions were obtained for each case study. The results show that a selective treatment of wastewater streams based on the search of local optimal solutions yields significant savings with respect to a centralized treatment design.

Modelo de Programación No Lineal para el Diseño de Sistemas de Tratamiento Electroquímico de Aguas Residuales Contaminadas con Trazas de Amoxicilina

In this work, a nonlinear programming (NLP) model is formulated for the design of selective effluent treatment systems containing traces of amoxicillin. The presence of antibiotic in surface water bodies or groundwater generates an important environmental problem due to its bio-accumulative potential and the development of antibiotic resilient bacteria. Current methods employed in the removal of drugs and antibiotics include chemical oxidation by Fenton reagents, flocculation and photochemical treatments. However, their removal efficiencies are still limited and studies are required for the optimization of the treatment costs. An electrochemical technology is incorporated in the system for the degradation of amoxicillin. The NLP model developed is built from a basic network superstructure that expands the space of alternative treatment possibilities, minimizing the total flowrate that is sent to treatment. Four illustrative examples are solved to show the versatility and usefulness of the model proposed.

Effect of Surface Contamination on the Drag of a Bubble Rising in a Line

The presence of surfactants critically increases the drag on bubbles rising in contaminated water compared with bubbles rising in pure water. This is explained by the Marangoni effect, occurring when the surface tension forces existing on the surface generate tangential shear stresses on the surface bubble. This mechanism has been studied by considering stagnant cap hypothesis to simulate the increase in the drag as a function of surface contamination. In this work, the steady drag for contaminated spherical bubbles was obtained numerically for \(0.1\le Re\le 200\) by using Comsol \({\text {Multiphysics}}^{\circledR }\) 3.5a assuming the stagnant cap hypothesis. The numerical values of the vorticity, flow velocity and pressure fields as function of the angle of superficial contamination and Re were examined. The agreement of the numerical results with reported drag values for clean and partially contaminated bubbles, as well as rigid spheres was proved. By using an appropriate normalization of the numerical data, a simple drag correlation for contaminated bubbles as a function of the spherical angle of the stiff surface zone was obtained.