Eotvos number

Eo = Eotvos number = gA dVc Re = Reynolds number = du /[L Ap = density difference between the phases p = density of continuous liquid phase d = drop diameter [L = continuous liquid viscosity surface tension u = relative velocity... 

In order to take into account the effect of surface tension and micro-channel hydraulic diameter, we have applied the Eotvos number Eo = g(pL — pG)d /(y. Eig-ure 6.40 shows the dependence of the Nu/Eo on the boiling number Bo, where Nu = hd /k] is the Nusselt number, h is the heat transfer coefficient, and k] is the thermal conductivity of fluid. All fluid properties are taken at the saturation temperature. This dependence can be approximated, with a standard deviation of 18%, by the relation ... 

The large heated wall temperature fluctuations are associated with the critical heat flux (CHE). The CHE phenomenon is different from that observed in a single channel of conventional size. A key difference between micro-channel heat sink and a single conventional channel is the amplification of the parallel channel instability prior to CHE. As the heat flux approached CHE, the parallel channel instability, which was moderate over a wide range of heat fluxes, became quite intense and should be associated with a maximum temperature fluctuation of the heated surface. The dimensionless experimental values of the heat transfer coefficient may be correlated using the Eotvos number and boiling number. 

The value of ki depends on the Eotvos number, Ed, and the Morton number, Gj, as shown in Figure 7.12. 

Plot ofk, against Eotvos number for slug flow... 

By assuming that the surface tension on the surface of a fluid sphere varied from the surfactant-free value, at the nose to zero at the rear, Savic also deduced a relationship between velocity and Eotvos number, shown in Fig. 3.7, which agrees qualitatively with the experimental results of Bond and Newton. Modifications of this approach for cases where the maximum change in local interfacial tension is less than have been devised for bubbles (D5, G7) and... 

Fig. 7.4 Reynolds number as a function of Eotvos number for systems with essentially identical M studied by different workers.

Equation (12-46) implies that the Eotvos number cannot exceed a value of about 16. Since the spherical-cap regime requires Eo > 40 (see Fig. 2.5), stability considerations explain why drops falling in gases and drops in many liquid-liquid systems never attain the spherical-cap regime. Moreover, since We = 4Eo/3Cd and is nearly constant for large drops in air, it is also possible to... 

For values of the Eotvos number higher than 40 (for the air-water system, this corresponds to dbub > 17 mm), the above equation simplifies to the well-known Davies-Taylor equation ... 

Static holdup is a function of the Eotvos number Eo (Van Swaaij el al., 1969) ... 

Figure 5.2-24. Experimental static liquid hold-up as a function of the Eotvos number (after Wammes et al. [34]).

Van Swaaij [52] and Charpentier et al. [53] proposed a relationship between the static liquid hold-up, I3stat, and the dimensionless Eotvos number, Eo. At high Eotvos numbers the static hold-up is inversely proportional to Eo, whereas at low Eotvos numbers, the static holdup reaches a maximum value. 

This correlation gives, for perfectly wettable solids, fairly good estimates of the static holdup for different particle-geometries and sizes. Saez and Carbonnel [26] used the hydraulic diameter, instead of the nominal particle diameter, as the characteristic length in the Eotvos number, to include the influence of the particle geometry on the static hold-up. However, no improvement could be obtained in correlating the data with this new representation. 

Wammes et al. [34], by employing three different liquids (water - ethanol and 40 % ethyleneglycol aqueous solution) with 3 mm glass spheres, obtained experimentally determined static hold-up data. Figure 5.2-24 shows the values of the static holdup as a function of the Eotvos number together with data of other authors. Wammes et al. [34] concluded that the static liquid hold-up is not affected by the total reactor pressure. 

Bond number Bo (Pi - Po)L2g gravitational force Atomization = Eotvos number, Eo... 

Gas Bubbles Fluid particles, unlike rigid solid particles, may undergo deformation and internal circulation. Figure 6-59 shows rise velocity data for air bubbles in stagnant water. In the figure, Eo = Eotvos number, g(pL - pG)dJa, where pL = liquid density pG = gas density, de = bubble diameter, a = surface tension, and the equivalent diameter de is the diameter of a sphere with volume equal to that of... 

The static holdup can be correlated with the Eotvos number NBo as it results from a balance of surface tension and gravity forces on the liquid held up in the pores in absence of flow ... 

For instance Fig. 19-42 illustrates the dependence of the static holdup on the Eotvos number for porous and nonporous packings. 

The static liquid holdup is often correlated by the Eotvos number, Eo (= pi.0dp/ffL> where dp is the nominal particle diameter and g the gravitational acceleration). Such a correlation103 is illustrated in Fig. 6-5. The correlation indicates that smaller particle diameter and fluid density and larger surface tension give larger static liquid holdup. The correlation also indicates that a porous material gives a larger static liquid holdup than a nonporous material. 

Wall thickness Channel width Acoustic velocity Friction coefficient Conductance Capillary number Discharge coefficient Drag coefficient Diameter Diameter Dean number Deformation rate tensor components Elastic modulus Energy dissipation rate Eotvos number Fanning friction factor Vortex shedding frequency Force... 

Atomization = Eotvos number, Eo Two-phase flows, free surface flows Compressible flow, hydraulic transients Cavitation... 

---