Weber number function

Static mixing of immiscible Hquids can provide exceUent enhancement of the interphase area for increasing mass-transfer rate. The drop size distribution is relatively narrow compared to agitated tanks. Three forces are known to influence the formation of drops in a static mixer shear stress, surface tension, and viscous stress in the dispersed phase. Dimensional analysis shows that the drop size of the dispersed phase is controUed by the Weber number. The average drop size, in a Kenics mixer is a function of Weber number We = df /a, and the ratio of dispersed to continuous-phase viscosities (Eig. 32). 

The first term is essentially the reciprocal of the Weber number and the second term is a function of the Ohnesorge number. Equation 13 may be invaHd for airblast atomizers operating at high pressures, >1 MPa (>10 atm), or with high viscosity Hquids. 

Clark and Vermeulen (C8) measured gas holdup in three different liquids —isopropyl alcohol, ethylene glycol, and water. They measured the increase in holdup with agitation as compared to no agitation, and correlated their results as a function of the volumetric gas velocity, Weber number, P/P0, and a geometric factor. Typical volumetric gas holdup values reported in the literature vary from about 2% to 40% of the total dispersion volume (Cl, C2, C8, F2, G10). 

Clark and Vermeulen (C8) later reported an extensive experimental study of power requirements in agitated gas-liquid systems. They correlated their data in dimensionless form as a function of fractional gas holdup, Weber number, and a geometrical factor. Their correlation is shown in Fig. 5. 

In their analysis, however, they neglected the surface tension and the diffusivity. As has already been pointed out, the volumetric mass-transfer coefficient is a function of the interfacial area, which will be strongly affected by the surface tension. The mass-transfer coefficient per unit area will be a function of the diffusivity. The omission of these two important factors, surface tension and diffusivity, even though they were held constant in Pavlu-shenko s work, can result in changes in the values of the exponents in Eq. (48). For example, the omission of the surface tension would eliminate the Weber number, and the omission of the diffusivity eliminates the Schmidt number. Since these numbers include variables that already appear in Eq. (48), the groups in this equation that also contain these same variables could end up with different values for the exponents. 

Senda et al)335 415 also derived equations describing the thickness and diameter of the radial film formed on a hot surface as a function of the Weber number, and correlated the mean diameter of droplets resulted from the breakup of the radial film with the thickness of the radial film and the Weber number. 

Pilcher (7S) describes the cascade impactor and makes a study of the dynamics of droplets. Stoker (12S) uses a surface coating of soot instead of magnesium oxide. He has found the impression diameter to be a function of the Weber number. Gillespie (2S)... 

Figure 3.5 The critical Weber number for disruption of droplets in simple shear flow (solid curve), and forthe resulting average droplet size in a colloid mill (hatched area) as a function ofthe viscosity ratio for disperse to continuous phases. Redrawn from data in Walstra [131].

Figure 3.6 Droplet breakup as a function of viscosity ratio. The solid line represents the critical Weber number value above which droplet breakup will occur. Data from Isaacs and Chow [130].

Figure 9.13 Critical Weber number for breaking up droplets as a function of the viscosity ratio, in pure shear flow and in extensional flow...

Fig.3 Density of the coexisting liquid phase Fig.4 Weber number of a falling film for the of the system oleic acid/carbon dioxide, oleic system oleic acid/ethane as a function of acid/ethane and pure oleic acid as fuction of ethane activity [4]. pressure [3].

The maximum observable particle size (x ax) is a function (eq.l) of the surface tension between continuous and disperse phase (y), the Weber number (We) and inversely proportional to the shear energy (t) introduced by the colloid mill. Small particles can be expected with high shear and small surface tension values [10-12]. 

The Weber number is the relation of the viscosities of the dispersed and continuous phase. It should be between 0,01 and 10. Interesting information can be gathered when the changes of the viscosities of dispersed and continuous phase are measured as a function of temperature. Finer droplets are often formed at lower temperatures. All three properties can be controlled most straightforward is the control of the shear energy. 

The predictive method for drop size is given in the Kenics Bulletin (May 1988, p. 28, Fig. 5-1) and in Figure 10.34. The ratio of Sauter mean drop size to the mixer ID (d/D) is a function of the Weber Number (V Dp/cr) and the ratio of dispersed phase to continuous phase viscosity (p-j/p.,). Now let s do two examples for static mixers. 

Cd, the drag coefficient, is a function of the Reynolds number Re) and the Weber number (We), both based on the bubble diameter (db) and slip velocity ( ml — An equation can be added for trans-... 

Multiphase reactor types are highly varied. The simplest approach to analyzing and predicting their behavior is to focus on the rate limiting steps or segment the reactor and model each segment and its contributions separately. Correlations are invariably a function of phase-based Reynolds and Froude numbers. Fractional volumes and properties of the solids are factors. Where interfacial tension is an important factor, the Weber number can be added. 

Figure 10-12. The shape of a rising gas bubble as a function of the Weber number, as predicted by small-deformation, boundary-layer theory. We = 0,0.25, 0.5.

The dependence of drop deformation on the Weber number and the vorticity inside the drop was studied in [336]. It was shown that the drop is close in shape to an oblate ellipsoid of revolution with semiaxis ratio > 1 If there is no vortex inside the drop, then this dependence complies with the function We(x) given in (2.8.3). The ratio x decreases as the intensity of the internal vortex increases. Therefore, the deformation of drops moving in gas is significantly smaller than that of bubbles at the same Weber number We. The vorticity inside an ellipsoidal drop, just as that of the Hill vortex, is proportional to the distance TZ from the symmetry axis,... 

Different assumption such as dj = d proposed by Schneider and Heinrichs are used to determine the size of the jet diameter. As reported by Whelehan and Marison, it was however demonstrated by Brandenberger and Widmer that for precision-drilled sapphire stone nozzles, the relationship between d and d is a function of the Weber number of the nozzle (We,. ... 

Stroeve and Varanasi (103) examined also the break up of the multiple-emulsion globules in a simple shear flow and concluded from the critical Weber number [(we), j.] (Figs. 35 and 36) that the multiple emulsion exhibits behavior that is similar to that of simple emulsions. From Fig. 35 one can see at least qualitatively, from the evolution of as a function ofp (the viscosity ratio between the... 

Figure 35 The critical Weber number for simple and multiple emulsions disruption as a function of the viscosity ratio dispersed to continuous phase. (From Ref 103.)...

Hinze interpreted data from Clay (31) in order to determine a value of 0.725 for C which allows the diameter in Couette flow to be deduced [see Eq. (4)]. Karabelas (32) questioned the assumptions made in Eq. (3), as turbulent flows are sometimes neither isotropic nor homogeneous. However, a number of workers have found the expression to be satisfactory. The dg diameter may also be expressed as a function of the Weber number ... 

The diameter of fluid particles is mainly a function of a Weber number which is composed of the local specific power input s or the square of the fluctuating velocity u f ... 

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