Granular temperature in a gas fluidized bed

The mean square of particle velocity fluctuations, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta v^{2}$$\end{document} , which is directly related to the so-called granular temperature, plays an important role in the flow, mixing, segregation and attrition phenomena of particulate systems and associated theories. It is, therefore, important to be able to measure this quantity. We report here in detail our use of diffusing wave spectroscopy (DWS) to measure the mean square particle velocity fluctuations for a 2D non-circulating gas fluidized bed of hollow glass particles whose mean diameter and effective density are 60 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\upmu$$\end{document} m and 200 kg/m3, respectively. Mean square particle velocity fluctuations were observed to increase with superficial velocity, Us, beyond the minimum fluidization velocity. Following the uniform fluidization theory of Batchelor (1988), the function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f{\left(\phi \right)}$$\end{document} in the expression \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta v^{2} = f{\left(\phi \right)}U^{2}_{\rm s}$$\end{document} was also determined and shown to increase from zero at a solids loading of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi \approx 0.33$$\end{document} to a maximum at \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi \approx 0.4$$\end{document} before decreasing again to zero at \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi \approx 0.53$$\end{document} . The spatial variation of the mean square particle velocity fluctuations was also determined and shown to be approximately symmetrical about the centreline where it is also maximal, and to increase with height above the distributor.