A Study of Spouts on Bath Surfaces from Gas Bubbling: Part II. Elucidation of Plume Dynamics
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TRODUCTION
THE study of fluid dynamics of gas-stirred ladles has been the subject of many articles in the literature. There are generally two categories of models: computational fluid mechanic models of gas and liquid flows, which calculate local values of phase fraction and velocity (reviewed by Mazumdar and Guthrie[1]); and macroscopic models that make balances over the plume and bulk flow regions in various ways to yield average quantities (reviewed by Mazumdar and Evans[2]). The present work is an original contribution in the second class of models that is simple and accurate enough to be used in design calculations. The objective of this work is to use the information from Part I[3] on spout heights to elucidate the fluid mechanics in the plume; these include the following: (1) the scaling factor for gas flow rates in gas-stirred systems, (2) the volume-averaged void fraction in the plume, (3) the cross-sectional area-averaged void fraction as a function of height, (4) the cross-sectional area-averaged liquid velocity in the plume as a function of height; and (5) the cross-sectional area-averaged gas velocity in the plume as a function of height. [3]
As well as using the Part I information, data from other works has been used to provide a unified macroscopic model of the plume dynamics, and the model is applicable to other unconfined plumes. It will also be proposed that spout height measurements made in more confined systems or multi-injection systems can be used to model the aforementioned phase properties. K. KRISHNAPISHARODY, Postdoctoral Fellow, and G.A. IRONS, Dofasco Professor of Ferrous Metallurgy and Director, are with the Steel Research Centre, McMaster University, L8S 4L7, Hamilton, ON, Canada. Contact e-mail: [email protected] Manuscript submitted May 19, 2006. Article published online June 27, 2007. METALLURGICAL AND MATERIALS TRANSACTIONS B
II.
SCALING RELATIONSHIP FOR GAS-STIRRED SYSTEMS
Small-scale water models are often used to simulate the flow in a full-scale metallurgical vessel, usually referred to as prototype. The models are usually geometrically similar to the prototype and are scaled down by a factor of k in linear dimensions. The scaling factor for the gas flow rate has not been determined conclusively; values of k1.5 (Mazumdar[4]), k2.5 (Mazumdar et al.[5] and Kim and Fruehan[6]), and k2.75 (Sahai and Guthrie[7]) have been proposed. In the present work, the concept of similarity is applied in the inverse manner that it is normally used. Normally, a system is scaled down by k, and the relevant dimensionless groups, such as the Froude number, are used to ensure dynamic similarity, so that the forces are in the same ratio in the model and full-scale system; this condition then fixes all dimensions and forces imposed by the system in the same ratio. In this work, the spout heights in the prototype and the model are fixed to be geometrically similar: hp ¼k hm
½1
where the subscript p is used for the prototype and m for the model. The experimentally derived expression for the spout height in Pa
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