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Experiments performed with bubbly flow in vertical pipes at different flow conditions covering the transition region : simulation by coupling Eulerian, Lagrangian and 3D random walks models

Munoz-Cobo J., Chiva S., Essa M., Mendes S.

Archives of Thermodynamics

2012

Vol. 33, no. 1

3-39

Two phase flow experiments with different superficial velocities of gas and water were performed in a vertical upward isothermal co-current air-water flow column with conditions ranging from bubbly flow, with very low void fraction, to transition flow with some cap and slug bubbles and void fractions around 25%. The superficial velocities of the liquid and the gas phases were varied from 0.5 to 3 m/s and from 0 to 0.6 m/s, respectively. Also to check the effect of changing the surface tension on the previous experiments small amounts of 1-butanol were added to the water. These amounts range from 9 to 75 ppm and change the surface tension. This study is interesting because in real cases the surface tension of the water diminishes with temperature, and with this kind of experiments we can study indirectly the effect of changing the temperature on the void fraction distribution. The following axial and radial distributions were measured in all these experiments: void fraction, interfacial area concentration, interfacial velocity, Sauter mean diameter and turbulence intensity. The range of values of the gas superficial velocities in these experiments covered the range from bubbly flow to the transition to cap/slug flow. Also with transition flow conditions we distinguish two groups of bubbles in the experiments, the small spherical bubbles and the cap/slug bubbles. Special interest was devoted to the transition region from bubbly to cap/slug flow; the goal was to understand the physical phenomena that take place during this transition A set of numerical simulations of some of these experiments for bubbly flow conditions has been performed by coupling a Lagrangian code, that tracks the three dimensional motion of the individual bubbles in cylindrical coordinates inside the field of the carrier liquid, to an Eulerian model that computes the magnitudes of continuous phase and to a 3D random walk model that takes on account the fluctuation in the velocity field of the carrier fluid that are seen by the bubbles due to turbulence fluctuations. Also we have included in the model the deformation that suffers the bubble when it touches the wall and it is compressed by the forces that pushes it toward the wall, provoking that the bubble rebound like a ball.

Nonstandard eulerian and hamiltonian graphs, and a coloring theorem for nonstandard graphs

Zemanian A. H.

Zeszyty Naukowe. Telekomunikacja i Elektronika / Akademia Techniczno-Rolnicza w Bydgoszczy

2006

z. 9

37-44

From any given sequence of finite or infinite graphs, a nonstandard graph can be constructed. The procedure is similar to an ultrapower construction of an internal set from a sequence of subsets of the real line, but now the individual entities are the nodes of the graphs instead of real numbers. The transfer principle can then be invoked to extend several graph-theoretic results to the nonstandard case. In this work, nonstandard versions of Eulerian graphs, Hamiltonian graphs, and a coloring theorem are established for nonstandard graphs.

Z dowolnej sekwencji grafów skończonych lub nieskończonych można zbudować graf niestandardowy. Procedura w tym przypadku jest podobna do konstrukcji tzw. ultramocy zbioru wewnętrznego z sekwencji podzbiorów zbioru liczb rzeczywistych, gdzie teraz poszczególne wielkości są węzłami grafów, a nie liczbami rzeczywistymi. Wtedy zasada transferu może być zastosowana w celu rozszerzenia zakresu stosowalności wielu teoretycznych wyników dotyczących grafów standardowych na grafy niestandardowe. W tej pracy są rozpatrywane niestandardowe przypadki grafów Eulerowskich i Hamiltonowskich oraz twierdzenie o kolorowaniu dla przypadku grafów niestandardowych.