Froude number for

Figure 3.9 compares the separated-to-intermittent flow regime boundaries for both 8-in.- and 4-in.-pipe experiments. The conditions are presented in terms of nondimensional superficial velocities J = Jk[pk/(pL - pc)gZ)]1/2 (or the density-modified Froude numbers). For 4-in.-pipe experiments, data have been obtained only for 3 MPa, while for the 8-in.-pipe experiments, data have been obtained for pressures of 3.0, 5.0, and 7.3 MPa. Note the value of J at the flow regime transition increased with pressure for the 8-in.-pipe experiments. 

The Froude number for mixing FrM represents the ratio of the applied to the opposing gravitational forces. 

It can be shown by dimensional analysis [Holland and Chapman] that the power number Po can be related to the Reynolds number for mixing ReM, and the Froude number for mixing FrM, by the equation... 

A plot of Po against ReM on log-log coordinates for the unbaffled system gives a family of curves at ReM > 300. Each curve has a constant Froude number for mixing FrM. 

For additional information, see Simpson (Chem. Eng., 75[6], 192-214 [1968]). A critical Froude number of 0.31 to ensure vented flow is widely cited. Recent results (Thorpe, 3d Int. Conf. Multi-phase Flow, The Hague, Netherlands, 18-20 May 1987, paper K2, and 4th Int. Conf. Multi-phase Flow, Nice, France, 19-21 June 1989, paper K4) show hysteresis, with different critical Froude numbers for flooding and unflooding of drain pipes, and the influence of end effects. Wallis, Crowley, and Hagi (Trans. ASME J. Fluids Eng., 405-413 [June 1977]) examine the conditions for horizontal discharge pipes to run full. 

It is decided to model a full-scale prototype, unbaffled, stirred vessel with a one-tenth scale model. The liquid in the prototype has a kinematic viscosity, v. of 10 7 m2 s As we have seen above, power number is a function of both Reynolds number and Froude number for unbaffled vessels. To ensure power number similarity, we need to ensure both Reynolds number and Froude number are similar from prototype to model. 

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. 

Froude Number for Collete-Gral High-Shear Mixers... 

Fig. 9 The range of Froude numbers for Collette Gral high-shear mixers.

The range of Froude numbers for Fielder PMA series mixers is shown on Fig. 10. The 10-L laboratory scale mixer at its lowest speed settings can reach the Froude numbers of all other mixers, except one. These considerations can be useful for planning a scale-up or technology transfer operation. 

The following equations will calculate Froude numbers for both the liquid and gas phases. The computer program will print out a message indicating whether the vertical pipe is self-venting, whether pulsating flow occurs, or whether no pressure gradient is required. 

The computer program PROG36 calculates the Froude numbers for the liquid and vapor phases. In addition, PROG36 will determine whether the pipe is self venting or whether pulsation flow is encountered. Table 3-13 shows the results for the 2-, 4-, and 6-inch (Schedule 40) pipes. Table 3-14 gives a typical input data and computer output for the 2-inch (Schedule 40) pipe. 

FORMAT (lOX, FROUDE NUMBER FOR LIQUID PHASE , T65, F8.4, /, lOX, FROUDE NUMBER FOR VAPOR PHASE , T65, F8.4)... 

The Froude number for the total flow flowing as liquid (Fr,(.) is given as... 

Fig. 5 Relationship between dimensionless mixing time and equivalent Froude number for the colunm type bioreactor...

White and de Villiers (1977) defined a modified Froude number. For N=N, ... 

Thorat et al. (2001) have given Equation 10.22 for the critical Froude number for bubble columns when the dispersion height is much larger than the maximum encountered in sieve trays ... 

Colwell and O Bara recommended applying equitation (12.53) for cases with Froude number less than 0.2 and equation (12.55) for large Froude numbers. For high pressure towers (>165psia), the Hsieh and McNulty (1986) correlation is recommended and details are provided there. 

D Mean particle size, mm F Bed frequency of oscillation, s Solids mass flow rate, kg/s Fr Froude number for slurry flow in the drum, V/[2 g R (Ss - 1) ] Fr, Modified Froude number for pipe... 

Fr Modified Froude number for open channel flow V /[g L (S - 1)] ... 

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