Characterization of materials by Hallimond tube flotation. Part 1: maximum size of entrained particles

Abstract

Bubbling gases through water in monobubble Hallimond tubes cause entrainment of fine particles. The particles are entrained with a layer of water traveling behind bubbles. Four modes of bubble movement in the monobubble Hallimond tube — the vertical coalescent string of bubbles, Archimedean spiral, symmetrical spiral, and lateral bubble movement in the inclined part of the Hallimond tube — were distinguished and discussed. Maximum vertical velocity of the bubbles in all four regions of the Hallimond tube was found to be within 13.1 ± 0.7 cm/sec. The Reynolds number of the bubbles produced in the monobubble Hallimond tube is between 385 and 947 indicating a near turbulent or turbulent flow in the layer of water behind the bubble. The particles traveling with the water layer behind bubbles are subjected to gravity which causes their settling. Particles which settle slower than the velocity of the water layer behind a bubble can be entrained with the bubbles and transferred to the receiver of the Hallimond tube. On the basis of this principle two equations for the maximum size of entrained particles ( d max) in the monobubble Hallimond tube were derived. The equation for particles which obey Newton's law of settling is: d max (ρ p−ρ 1) ρ l≅L H = 3( w b O ) 2ζ 4 g ( cm) while for particles which settle according to Allen's law, the equation is: d max (ρ p−ρ 1) ρ l ) 0.75≅L L= w b O 113.2 ( cm) where ϱ p is the density of the particles (g/cm 3), ϱ 1 density of the liquid (g/cm 3), w b o vertical velocity of a bubble within the slowest region of the Hallimond tube (cm/s), ζ drag coefficient (dimensionless number whose value depends on Reynolds number), g acceleration due to gravity (cm/s 2), and L h and L l are constants. Experimental tests carried out in a monobubble Hallimond tube using 13 different hydrophilic materials confirmed the applicability of formula (a) for materials with densities above 2.0 g/cm 3 and formula (b) for lighter substances. This was so, since the following experimental formulae were obtained for regularly shaped particles: max (ρp−ρ 1 ρ l =L H=0.0023±0.002( cm)( forρ≥2.0g/ cm 3) and max (ρp−ρ 1 ρ l) 0.75 =L L=0.0020±0.002( cm)( forρ≤2.0g/ cm 3) However, the theoretical value of L h was 2.5 and L l 5.5-fold greater than that obtained experimentally. Factors which may be responsible for the discrepancy were indicated.