Dimensionless number Weber

F), in addition to the Reynolds and Weber numbers, to fully describe a droplet spreading and solidification process upon impact on a substrate. They introduced two new dimensionless numbers, defined as ... 

The letters R, F, and W stand for so-called Reynolds, Froude, and Weber numbers, respectively these are dimensionless numbers, as indicated. For example, if we make the Reynolds number the same in model and prototype, using the same fluid, the dimension of length is smaller in the model and hence the velocity v will have to be greater. In other words, the water would have to flow faster in the model. If we now consider the Froude number as the same in model and prototype, and that the same fluid is used in both, we see that the velocity would have to be less in the model than in the prototype. This may be regarded as two contradictory demands on the model. Theoretically, by using a different fluid in the model (thus changing p0 and p), it is possible to eliminate the difficulty. The root of the difficulty is the fact that the numbers are derived for two entirely different kinds of flow. In a fluid system without a free surface, dynamic similarity requires only that the Reynolds number be the same in model and prototype the Froude number does not enter into the problem. If we consider the flow in an open channel, then the Froude number must be the same in model and prototype. 

Nsc Schmidt number (= p/pD), dimensionless Nsh Sherwood number (= kEdp/D around bubbles) (= ksdp/D around particles), dimensionless Nwe Weber number, pe (o2dx/a, dimensionless p pressure in reactor, N/m2... 

The creation of spherical droplets has been the focus of numerous studies over the years [11-14]. These show conclusively that for both steady and transient flows the onset and mechanisms of droplet breakup can be correlated with the non-dimensional Weber number. It is the most important dimensionless number characterizing droplet formation and can be applied to determine the threshold of droplet formation. However, the critical Weber number is only a sufficient condition for droplet breakup and not a necessary condition. This means that if the critical Weber number is surpassed in a process certainly droplet breakup will occur. But droplet ejection is also possible at lower Weber numbers. The only necessary condition for droplet formation is that the supplied energy is sufficient to overcome friction losses and the surface energy of an ejected droplet. 

The Weber number is the most important dimensionless number characterizing droplet formation. Therefore, the Weber number can be used to distinguish between occurring droplet formation regimes. 

In addition to the dimensionless numbers, there are well-known others, such as the Sherwood (Sh), Reynolds (Re), Schmidt (Sc), Froude (Fr), Bodenstein (Bo), and Weber (We) numbers. On the basis of these types of dimensionless numbers, empirical correlations for a large number of bioreactors have been made (for example, Blanch, 1979 Schiigerl, 1980 Zlokarnik, 1979). The results of the experimental measurements of process engineering data are often presented in the form of a graph they have the form of the relationships given in Equs. 3.77a and 3.77b. For the volumetric mass transport coefficient (Ryu and Humphrey, 1972) (see Fig. 3.21)... 

In turbulent flow, droplets are deformed and disrupted mostly by inertial forces that are generated by energy-dissipating small eddies. Due to internal viscous forces, droplets try to regain their initial form and size [ 12]. Two-dimensionless numbers, the turbulent Weber number Wfcturb and the Ohnsorge number Oh, characterize the tensions working on droplets in deformation and break-up ... 

Droplet dispensing is the procedure of ejecting single droplets or small jets out of a nozzle of a dispensing apparatus. Here we only consider liquid droplet dispensing into a gaseous environment. Dimensionless numbers like the Reynolds, the Weber and the Ohnesorge numbers are well suited to describe the droplet formation qualitatively. 

Weber number (dimensionless) critical Weber number (dimensionless)... 

Eor air-assisted atomizer nozzles the interaction of the gas and liquid phases, gas-liquid ratio GLR, and total mass flow rate on the different levels of structure of SEs and DEs has been discussed. The effect of the disperse to continuous viscosity ratio A on the dispersion of emulsions in spray processing has been extensively explored, and a new dimensionless gas Weber number, Weg oropM related to the secondary emulsion drop diameter was defined. The resistance of the secondary emulsion droplets in two-phase nozzle spray processing (air assist) was given up to a critical value (Weg,DropA)c-... 

The flow in fluid-fluid microstructured channels is characterized using dimensionless numbers. The most important dimensionless number for characterization of all types of flows is the Re number that relates inertial force to viscous force. Due to low flow velocities and characteristic dimension in the micrometer range, Re is often less than 1 meaning that viscous force is dominant over inertial force. The capillary number Ca is the ratio of viscous to interfacial forces. The range of Ca in a typical microchannel is lO " to 10 . Multiplying both numbers. Re and Ca, results in the Weber number We, which represents the ratio between inertial and interfacial forces. The importance of gravity vhth respect to interfacial forces is characterized by the Bond number Bo. The definitions of the dimensionless numbers are summarized in Table 2.2. 

The hydrodynamics in liquid-liquid MSR is mainly governed by interfacial tension, inertial forces, viscous forces, and density and viscosity ratios. However, the transition of flow pattern boundary is a weak function of the last two ratios, as long as these ratios remain much smaller than unity [36]. The first three forces can be combined in the form of dimensionless numbers such as Reynolds, capillary, and Weber numbers. 

Weber number A dimensionless number, We, used in thin film gas-liquid flows where surface tension has a major effect on the strongly curved interface between two fluids such as in the formation of droplets and bubbles. It is a measure of the relation between surface... 

Fig. 32. Dimensionless drop size vs Weber number A, empty pipe at pu j = 1 B through G, Kenics mixer at = 25, 10, 2, 1, 0.75, and 0.5,...

Based on such analyses, the Reynolds and Weber numbers are considered the most important dimensionless groups describing the spray characteristics. The Reynolds number. Re, represents the ratio of inertial forces to viscous drag forces. 

The Weber number becomes important at conditions of high relative velocity between the injected Hquid and surrounding gas. Other dimensionless parameters, such as the Ohnesorge ((We /Re), Euler (AP/Pj y i)y and Taylor (Re/ We) numbers, have also been used to correlate spray characteristics. These parameters, however, are not used as often as the Reynolds and Weber numbers. 

Impeller Weber number, Dimensionless Dimensionless decanter m ... 

Both dimensionless Weber number and viscosity ratio are defined by ... 

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. 

The first dimensionless group on the right is the Reynolds number, the second represents the ratio of the gas velocity to the impeller tip speed, the third is the Weber number, and the fourth is the Froude number. 

Pattern transition in horizontal adiabatic flow. An accurate analysis of pattern transitions on the basis of prevailing force(s) with flows in horizontal channels was performed and reported by Taitel and Dukler (1976b). In addition to the Froude and Weber numbers, other dimensionless groups used are... 

Table 12.1 gives a summary of the dimensionless variables. Two additional groups have been added, the Weber number, We, to account for droplet formation and the Nusselt number, Nu = hj/k, to account for gas phase convection. A corresponding Nusselt... 

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