Dimensionless groups tension

Flow Past Deformable Bodies. The flow of fluids past deformable surfaces is often important, eg, contact of Hquids with gas bubbles or with drops of another Hquid. Proper description of the flow must allow for both the deformation of these bodies from their shapes in the absence of flow and for the internal circulations that may be set up within the drops or bubbles in response to the external flow. DeformabiUty is related to the interfacial tension and density difference between the phases internal circulation is related to the drop viscosity. A proper description of the flow involves not only the Reynolds number, dFp/p., but also other dimensionless groups, eg, the viscosity ratio, 1 /p En tvos number (En ), Api5 /o and the Morton number (Mo),giJ.iAp/plG (6). 

Where surface-active agents are present, the notion of surface tension and the description of the phenomena become more complex. As fluid flows past a circulating drop (bubble), fresh surface is created continuously at the nose of the drop. This fresh surface can have a different concentration of agent, hence a different surface tension, from the surface further downstream that was created earlier. Neither of these values need equal the surface tension developed in a static, equiUbrium situation. A proper description of the flow under these circumstances involves additional dimensionless groups related to the concentrations and diffusivities of the surface-active agents. 

Roy et al. (R3) define the critical solids holdup as the maximum quantity of solids that can be held in suspension in an agitated liquid. They present measurements of this factor for various values of gas velocity, gas distribution, solid-particle size, liquid surface tension, liquid viscosity, and a solid-liquid wettability parameter, and they propose the following two correlations in terms of dimensionless groups containing these parameters ... 

Larachi et al. [37] presented a simplified version of Ellman s correlation. A friction factor, fiGG, is represented as a function of dimensionless groups which takes inertia, viscosity and surface-tension effects into account by using, respectively, %g, Rei, and Wee. 

Single screw extruder. Let us take the case of a single screw extruder section that works well when dispersing a liquid additive within a polymer matrix. The single screw extruder was already discussed in the previous section. However, the effect of surface tension, which is important in dispersive mixing, was not included in that analysis. Hence, if we also add surface tension as a relevant physical quantity, it would add one more column on the dimensional matrix. To find the additional dimensionless group associated with surface tension, as, and size of the dispersed phase, R, two new columns to the matrix in eqn. (4.32) must be added resulting in ... 

A common way to produce small droplets is to let them drip slowly from a capillary. The the size of the droplet, Dd, depends on surface tension, as, the diameter of the capillary, D, gravity, g, and density of the liquid. Perform a dimensional analysis to determine the dimensionless groups that govern this process. 

In Eq. (48), y is the coefficient of surface tension, g is gravitational acceleration and Apm is the difference in mass densities between the aqueous and organic liquids. The interface position z = (r) and the deflection t(r) = — of the interface from its unperturbed position are shown schematically in Fig. 6. Nondimensionalization of Eq. (48) leads to two dimensionless groups that relate electrostatic and gravitational stresses to surface tension. These groups are called the electrostatic and gravitational bond numbers, and are given by [25]... 

In Table II, we list the various dimensionless groups defined above and the two criteria for enhancement of the heat transfer coefficient by interfacial tension-driven flow. Calculated values are given for the three mixtures for which we presented experimental data. All values pertain to a temperature difference AT of 10 K. 

Predicted Deformation Mechanisms. Recent work has developed maps of the deformation mechanisms expected in films with different properties. Two dimensionless groups were found to determine which of the deformation mechanism occurs. The first is the time for particle deformation compared to the time for evaporation, captured in 1 = ERt]o/yH, where E is the evaporation rate, t]o is the polymer viscosity, and y is the water-air surface tension. The second dimensionless group is the Peclet number, which determines the vertical homogeneity in the film, Pe = 6nt] R H E/kT. The deformation regimes are shown in Fig. 9. 

Here, Fv and Fc are viscous and capillary forces, respectively v is the pore flow velocity of the displacing fluid in their derivation p is the displacing fluid viscosity and a is the interfacial tension between the displacing and displaced phases. The dimensionless group is called a capillary number, Nc. A set of consistent units is used so that the dimensionless group is dimensionless. For example, v is in m/s, p in mPa s, and a in mN/m or dyne/cm. 

However, if each of the films is considered separately, two equations similar to Eq. (1), but excluding the diffusivity-ratio term, are obtained (H5). Determination of the functional relationship between the various dimensionless groups is quite difficult, and many of the correlations reported (see Tables I-III) are abbreviated forms of Eq. (1). Obviously these introduce the assumption that effects of interfacial tension, phase viscosities, and density difference on the transfer coefficient are constant over the usual range of these properties. Since the range is normally narrow, the assumption is quite reasonable, though one may expect a fairly wide scatter of measured values for unusual systems. 

Within the bubble boiling regime, thermal induced disintegration occurs when the vapor pressure unbalances the equilibrium between surface tension, viscous forces and inertial forces. The nature of this mechanism is different from those observed onto cold surfaces, as it is triggered by combined effects induced by the liquid surface tensirm and the latent heat of evaporation, /ifg, and the analysis requires the use of dimensionless groups complementary to those in Table 8.1. The most important is the Jakob number, defined as/a = Cp(Tw — 7 sat)//tfg where Cp is the specific heat of the liquid. 

Table 4.1 presents the dimensionless groups that are important in jet flow and jet break up times, where R is the nozzle radius and Uq is the velocity at the nozzle, p, p and o are the density, viscosity and surface tension of the ink. 

Generally, phase morphologies produced in blending involve disperse phases sizes that vary with interfacial tension, k, or with the dimensionless group Kjrjv or K/cTi2d, where is viscosity, vis velocity, and(T12 a shear stress [154,155]. This dimensionless group represents a ratio of interfacial to viscous forces. 

One may interpret this dimensionless group ijay/K (or ijaE/K) as the ratio of viscous forces tending to deform (and break up) droplets to the interfacial tension forces tending to hold them together. It is most appropriately called a Taylor number. 

This dimensionless grouping represents the competitive interaction of the viscous driving forces and the restrictive capillary forces. High recovery efficiencies, > 90%, have been realized for laboratory systems in which the oil-water interfacial tension has been reduced to 10"2 - 10" dynes/cm. Such lowered tensions are difficult to establish and sustain under field conditions. Recently, more attention has been placed on mechanisms which alter the configuration of the residual phase (1,2). 

These dimensionless groups give the relative importance of gravitational, viscous and inertial forces to the surface tension force, respectively. Note that the values of all of these numbers decrease significantly as the system size diminishes (small diameters and small velocities), increasing the importance of surface tension effects. 

For nonspherical particles such as ellipsoids, two particle-size variables are needed, such as the major and minor semiaxes a and b. Superpositions can be expected for systems of comparable values of the dimensionless axial ratio r = alb. For deformable solid particles, the elastic modulus G governs deformation under shear this requires a new dimensionless group such as the ratio a/G. For emulsions, both the viscosity of the particle and the interfacial tension F will influence rheological behavior. The new dimensionless groups are the viscosity ratio and the stress ratio cta/F. Systems of interacting particles will be characterized by... 

The flow can be forced, e.g., by a pump or an impeller, it can be driven by gravity or even by surface tension gradients. In most cases, the flow is too complex to be analysed theoretically, and empirical correlations are used for predicting values of the mass transfer coefficient for given conditions. These correlations are usually presented as functions of dimensionless groups, in order to generalize the applicability. 

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