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13.8.2: Solid With Lighter Density \(\rho_S< \rho\) and With Gravity

This situation is minimal and very few cases exist. However, it must be pointed out that even in solid–gas, the fluid density can be higher than the solid (especially with micro gravity). There was very little investigations and known about the solid–liquid flowing down (with the gravity). Furthermore, there is very little knowledge about the solid–liquid when the solid density is smaller than the liquid density. There is no known flow map for this kind of flow that this author is aware of. Nevertheless, several conclusions and/or expectations can be drawn. The issue of minimum terminal velocity is not exist and therefor there is no fixed or mixed fluidized bed. The flow is fully fluidized for any liquid flow rate. The flow can have slug flow but more likely will be in fast Fluidization regime. The forces that act on the spherical particle are the buoyancy force and drag force. The buoyancy is accelerating the particle and drag force are reducing the speed as
    \dfrac{\pi\,D^3\,g(\rho_S - \rho_L)}{6} =
    \dfrac{{C_D}_\infty\,\pi \, D^2 \rho_L \left( U_S-U_L\right)^2 }{8}
     \label{phase:eq:ligthSolid}  \tag{54}
From equation 13.54, it can observed that increase of the liquid velocity will increase the solid particle velocity at the same amount. Thus, for large velocity of the fluid it can be observed that \(U_L/U_S \rightarrow 1\). However, for a small fluid velocity the velocity ratio is very large, \(U_L/U_S \rightarrow 0\). The affective body force "seems'' by the particles can be in some cases larger than the gravity. The flow regimes will be similar but the transition will be in different points. The solid–liquid horizontal flow has some similarity to horizontal gas–liquid flow. Initially the solid particles will be carried by the liquid to the top. When the liquid velocity increase and became turbulent, some of the particles enter into the liquid core. Further increase of the liquid velocity appear as somewhat similar to slug flow. However, this author have not seen any evidence that show the annular flow does not appear in solid–liquid flow.


  • Dr. Genick Bar-Meir. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or later or Potto license.