Dimensionless groups Weber number

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 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... 

Each dimensionless group represents a rule for scale-up. Frequently these individual scale-up rules conflict. For example, scale-up on dynamic similarity should depend chiefly upon a single dimensionless group that represents the ratio of the applied to the opposing forces. The Reynolds, Froude and Weber numbers are the ratios of the applied to the resisting viscous, gravitational and surfaces forces, respectively. 

In previous paragraphs, we obtained groups of dimensionless vaiiables by means of Buckingham s theorem. In the development of Reynolds, Froude, and Weber numbers we utilized the concept of force ratios although the same numbers can be produced by means of Buckingham s theorem. 

The value of dimensionless groups has long been recognised. As early as 1873, Von Helmholtz derived groups now called the Reynolds and Froude "numbers", although Weber (1919) was the first to name these numerics. 

For low viscosity liquids when the dimensionless viscosity group [7], NJ, = /LtiAcrp,) is less than 1, a droplet wiU be stable below a maximum size, defined by the critical Weber number and the gas liquid contact time. For long gas—liquid contact times, two large droplets are produced. For short contact times, many smaU droplets are produced. This tjq) of droplet breakup yields a very broad droplet size distribution. 

Governing equations are the continuity equation, the chemical reactions and their thermodynamic relationships, and the heat, mass, and momentum equations. Elastic behavior of an expanding bed of particles sometimes must be included. These equations can be many and complex because we are dealing with both multiphase and multicomponent systems. Correlations are often in terms of phase-based dimensionless groups such as Reynolds numbers, Froude numbers, and Weber numbers. 

There is a single dimensionless group, XVjL, which is known as the Weissenberg number, denoted by various authors as We or Wi. (We is more common, but it can lead to confusion with the Weber number, so Wi will be used here.) The shear rate in any viscometric flow is equal to a constant multiplied by V/L, so it readily follows that the ratio of the first normal stress difference to the shear stress is equal to twice that constant multiphed by Wi. Hence, Wi can be interpreted as the relative magnitude of elastic (normal) stresses to shear stresses in a viscometric flow. The ratio of the shear stress to the shear modulus, G, is sometimes known as the recoverable shear and is denoted Sr. Sr differs from Wi for a Maxwell fluid only by the constant that multiplies F jL to form the shear rate for a given flow. In fact, many authors define Wi as the product of the relaxation time and the shear rate, in which case Wi = Sr. It is important to keep the various definitions of Wi in mind when comparing results from different authors. 

---