Froude number significance

The good agreement obtained for all data using the modified Froude number signifies the physical significance of the parameter. In fact, the dependence of jet penetration on the two-phase Froude number can be derived theoretically from the buoyancy theory following that of Turner (1973). 

From equation (4.26), it is seen that the important dimensionless parameters driving momentum transport are the Reynolds number, the Froude number, the Euler number, and the length and velocity ratios in the flow field. The dimensionless variables all vary between zero and a value close to one, so they are not significant in determining which terms in the governing equation are important. 

Consider a fluidized bed operated at an elevated temperature, e.g. 800°C, and under atmospheric pressure with ah. The scale model is to be operated with air at ambient temperature and pressure. The fluid density and viscosity will be significantly different for these two conditions, e.g. the gas density of the cold bed is 3.5 times the density of the hot bed. In order to maintain a constant ratio of particle-to-fluid density, the density of the solid particles in the cold bed must be 3.5 times that in the hot bed. As long as the solid density is set, the Archimedes number and the Froude number are used to determine the particle diameter and the superficial velocity of the model, respectively. It is important to note at this point that the rale of similarity requires the two beds to be geometrically similar in construction with identical normalized size distributions and sphericity. It is easy to prove that the length scales (Z, D) of the ambient temperature model are much lower than those in the hot bed. Thus, an ambient bed of modest size can simulate a rather large hot bed under atmospheric pressure. 

This definition of 2VFr is of particular physical significance in film flow, since here the Froude number denotes the ratio of the mean film velocity to the celerity of a gravity wave in shallow liquid. (Wave celerity is defined as the velocity of a wave relative to the liquid on which the wave propagates.) The Weber number is defined by analogy. 

Early investigations were concerned mostly with physical properties, somewhat with particle characteristics but little with particle group behavior. Even so, significant results were obtained. For instance, the distinction between L/S fluidization and G/S fluidization, viz., particulate and aggregative, and the provision of criteria for such distinction (Wilhelm and Kwauk, 1948 Harrison et ai, 1961 Romero and Johanson, 1962), most of which were based on the Froude number. Other criteria were then proposed involving fluctuating parameters in fluidization (Rietema, 1967), for instance, pressure drop or voidage. 

Below the above-mentioned Re range, however, significant differences were found also in the turbulent range between both bubble movements. The Re values, above which the horizontal and vertical bubble streams were held in equilibrium, could be predicted with a turbulent Froude number. 

The Froude number Np, is a measure of the ratio of the inertial stress to the gravitational force per unit area acting on the fluid. It appears in fluid-dynamic situations where there is significant wave motion on a liquid surface. It is especially important in ship design. 

The group V l gz ) is dimensionless and is called the Froude number after William Froude (1810-1879). Its significance is discussed in Chap. 13. 

The measurements and predictions of friction factor vs. Re>Tiolds number with 100, 50 and 0 ppm solutions are shown in Fig. 1. The calculated results show good agreements with the measured values in the fully developed flow region. For different concentrations and different slopes, the same trend exists. No significant effect of the Froude number was apparent. 

One encounters the Reynolds number throughout chemical engineering. We will return later to the physical significance of the Froude number and the Reynolds number. 

Tables 15-2 to 15-5 show different parameters of significance for agitator mixers. Table 15-2 shows different classes in batch mixers followed by the mode of operation and Froude number and respective capacity ranges and their power requirements. Note that only the high intensity mixers have Fr 1, and centrifugal types with Fr > 1 otherwise, most of the mixers operate with Fr < 1. Sections to follow will treat each of the classes of mixers above.

At higher Froude numbers (>3), the mixing time is linear with mixer volume (not the diameter). The effect of agitator speed is significant in this range. 

At higher Froude numbers, the configuration of paddles/agitators will have a significant influence on the shape of the curve (Figure 15-57). The surface roughness, shape, and size of the particles also have a significant influence on this curve. 

For two-phase dispersions, other groups such as the iVeber number We = pu dp/ag, the ratio of inertial to surface forces, may be significant. Sherwood and Schmidt numbers are of course important in mass transfer. For dynamic similarity in two sizes of vessels operated with a vortex, the Reynolds and Froude numbers must be the same for both vessels. Since the impellers would be geometrically similar but unequally sized, it becomes impossible to specify the impeller speeds in the two vessels containing the same liquid to accomplish this. Thus, equal Reynolds numbers require... 

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