Dimensionless number Froude

This chapter reviews the various types of impellers, die flow patterns generated by diese agitators, correlation of die dimensionless parameters (i.e., Reynolds number, Froude number, and Power number), scale-up of mixers, heat transfer coefficients of jacketed agitated vessels, and die time required for heating or cooling diese vessels. 

Other dimensionless numbers are widely used for various scale-up applications. One example is Froude number ... 

Both methods yield dimensionless groups, which correspond to dimensionless numbers (1), e.g.. Re, Reynolds number Fr, Froude number Nu, Nusselt number Sh, Sherwood number Sc, Schmidt number etc. (2). The classical principle of similarity can then be expressed by an equation of the form ... 

Dimensionless numbers, such as Reynolds and Froude numbers, are frequently used to describe mixing processes. Chemical engineers are routinely concerned with problems of water-air or fluid mixing in vessels equipped with turbine stirrers where scale-up factors can be up to 1 70 (3). This approach has been applied to pharmaceutical granulation since the early work of Hans Leuenberger in 1982 (4). 

Hydrodynamic dimensionless numbers Examples are the Reynolds number, Froude, Archimedes, and Euler number. These dimensionless numbers have to be functions of identical determining dimensionless numbers of the same powers and with the same value of the other constant coefficients, so that the model and the object are similar. 

Gliksman s approach The result of the conversion of equations into nondimensional ones is a set of dimensionless parameters (Froude number, velocity, particle size, diameter ratios, etc.) that should be matched in both small and large systems. It is not necessary for the values of the parameters to be equal in each system. Instead, the dimensionless number ratios have to remain the same. To achieve this, the particle size and/or the particle density of the solids have to be changed appropriately in the small unit. It usually results in a smaller particle size in the small unit compared to the large one. 

A dimensional analysis of the system will result in four dimensionless numbers, Reynolds number, Froude number and geometric dimensionless parameters given by,... 

Densiometric froude a dimensionless number that describes the ratio of nonlinear advection to the pressure gradient acceleration associated with the variation of fluid depth. 

Richardson number dimensionless number that expresses the ratio of potential to kinetic energy, also known as the Froude number. 

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. 

Because the Reynolds number Re has already been taken as a governing dimensionless number, 7t7 will be taken as U2/gD i.e., as the Froude number, Fr. In some ways the use of U in this dimensionless number contradicts the use of V as a prime variable. However, this is not really a problem because the gravitational acceleration can only affect the heat transfer if there is a flow, i.e., if U is nonzero. Hence ... 

Another equipment parameter is the Froude number (Fr), a dimensionless number calculated as shown in Eq. (2), where v represents the impeller speed (rpms) and r is the radius of the granulator bowl. 

Dimensionless numbers most commonly used to describe the wet granulation process are Newton, Froude, and Reynolds ... 

Horsthuis et al. showed that an end-point could be reproduced and scaled up in Gral mixers by keeping the Froude numbers constant. For the same end-point, in dynamically similar mixers (same geometrical ratios, and same flow patterns), all dimensionless numbers describing the system should have the same numerical value, but Froude numbers for any mixer are easiest to compute. 

Scale-up in fixed bowl mixer-granulators has been studied by Rowe and Cliffs group using the classical dimensionless numbers of Newton (power), Reynolds, and Froude to predict the end-point in geometrically similar high-shear Fielder PMA 25, 100, and 600 L machines. 

The dimensionless source term essentially represents the ratio of generation to convection. For various generation terms, several additional dimensionless numbers may be defined. For example, if the generation of momentum due to gravitational forces is considered, a dimensionless number, called as the Froude number (Fr), is defined as the ratio of convection to gravitational factors. The dimensionless numbers discussed here along with other dimensionless numbers are listed in Table 2.1 together with their physical interpretation. 

Later Mayle, 1970 [400] continued their research by performing measurements of velocity and pressure within the fire whirl. He found that the behavior of the plume was governed by dimensionless plume Froude, Rossby, second Damkohler Mixing Coefficient and Reaction Rate numbers. For plumes with a Rossby number less than one the plume is found to have a rapid rate of plume expansion with height. This phenomenon is sometimes called vortex breakdown , and it is a hydraulic jump like phenomena caused by the movement of surface waves up the surface of the fire plume that are greater than the speed of the fluid velocity. Unfortunately, even improved entrainment rate type models do not predict these phenomena very well. 

The transitions between the various regimes generated by a gassed disk turbine can be quantified using dimensionless numbers such as the gas flow number, FIq(=Qg/ND ), the impeller Froude number, Fr (=N D/g), and system geometry ratios, such as the impeller diameter to tank diameter ratio (D/T)P For 6-blade disk turbines the cavity regime as well as other regimes are best obtained from the flow... 

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. 

For convenience, the relevant dimensionless numbers for gas-particle flow derived in this section are collected in Table 1.1. In practice, one must choose appropriate values for U and L corresponding to a particular problem. For example, they may be determined by the inlet and/or boundary conditions. However, one case of particular interest is particles falling in an unbounded domain for which convenient choices are T = t/p and U = ul = Up - f/gl = Tp g (i.e. the settling velocity). For this case, there is no source term for p and so it relaxes to zero at steady state due to the drag. The disperse-phase Mach number thus becomes infinite. For settling problems, the particle Archimedes number (see Table 1.1) is often used in place of the Froude number. 

The exact scale-up of crystallizers is not possible because it would be necessary to preserve similar flow characteristics of both liquid and solid phases together with identical temperatures and supersaturations in all equivalent regions. The scale-up of simple agitated vessels containing a liquid phase alone has long been recognized as a difficult problem. The two dimensionless numbers most frequently encountered in the analysis of stirrers and agitators are the Reynolds number, Re, and the Froude number, Fr ... 

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)... 

The Froude number is a dimensionless number that measures resistance. The greater the Froude number, the greater the resistance. 

Froude Number Dimensionless number for open-channel flow, defined as the ratio of the velocily to the square root of the product of the hydraulic depth and the gravitational acceleration. The Froude number equals 1 for critical flow. 

One practical approach during scale-up is to keep the processes dynamically similar, which is defined by the same dimensionless numbers for the two different sizes of granulators. Therefore, dimensionless numbers such as Froude number can be used for scale-up. 

---