Galileo number

The LHS of this expression contains a dimensionless group known as the Galileo number, defined as Ga = gdVv. ... 

The ratio Ap/p can be replaced by AT/T. Ar relates the influence of velocity and temperature of a jet when discharged into an environment of a different temperature. In some instances the Froude number, Galileo number, or Grashof number may replace the Archimedes number. 

Froude number, U2/gdp = Galileo number = Gravitational constant... 

As an alternative, the data used for the generation of equation 3.13 for the relation between drag coefficient and particle Reynolds number may be expressed as an explicit relation between Re 0 (the value of Re at the terminal falling condition of the particle) and the Galileo number Ga. The equation takes the form l0 ... 

The Galileo number is readily calculated from the properties of the particle and the fluid, and the corresponding value of Re 0, from which uq can be found, is evaluated from equation 3.40. 

Experimental results generally confirm the validity of equation 5.80 over these ranges, with n 4.8 at low Reynolds numbers and 2.4 at high values. Equation 5.78 is to be preferred to equation 5.79 as the Galileo number can be calculated directly from the properties of the particles and of the fluid, whereas equation 5.79 necessitates the calculation of the terminal falling velocity u0. 

More recently Khan and Richardson" 0 have examined the published experimental results for both sedimentation and fluidisation of uniform spherical particles and recommend the following equation from which n may be calculated in terms of both the Galileo number Ga and the particle to vessel diameter ratio d/dt ... 

Values of n from equation 5.84 are plotted against Galileo number for d/d, =0 and d/dt = 0.1 in Figure 5.17. It may be noted that the value of n is critically dependent on d/dt in the intermediate range of Galileo number, but is relatively insensitive at the two extremities of the curves where n attains a constant value of about 4.8 at low values of Ga and about 2.4 at high values. It is seen that n becomes independent of Ga in these regions, as predicted by dimensional analysis and by equation 5.80. 

Figure 5.17. Sedimentation index n as function of Galileo number Ga (from equation 5.84)...

The minimum fluidising velocity, umf, may be expressed in terms of the free-falling velocity o of the particles in the fluid. The Ergun equation (equation 6.11) relates the Galileo number Ga to the Reynolds number Re mj in terms of the voidage < , / at the incipient fluidisation point. 

In Chapter 3, relations are given that permit the calculation of Re 0(uodp/p), the particle Reynolds number for a sphere at its terminal falling velocity n0, also as a function of Galileo number. Thus, it is possible to express Re mp in terms of Re 0 and u ,f in terms Of Uq. 

For a spherical particle the Reynolds number Re 0 is expressed in terms of the Galileo number Ga by equation 3.40 which covers the whole range of values of Re of interest. This takes the form ... 

Figure 6.4. Ratio of terminal falling velocity to minimum fluidising velocity, as a function of Galileo number... 

Equation 6.31 is similar to equation 5.71 for a sedimenting suspension. Values of the index n range from 2.4 to 4.8 and are the same for sedimentation and for fluidisation at a given value of the Galileo number Ga. These may be calculated from equation 6.32, which is identical to equation 5.84 in Chapter 5 ... 

From Figure 6.4 it is seen that for emf = 0.40, <0/um/ is about 78 at low values of the Galileo number and about 9 for high values. In the first case, the drag on the particle is directly proportional to velocity and in the latter case proportional to the square of the velocity. Thus the force on a particle in a fluidised bed of voidage 0.4 is about 80 times that on an isolated particle for the same velocity. 

This is a quadratic equation in Re f where the Galileo number Ga is defined by... 

Richardson (1971) summarises a mefhod of predicting minimum fluidizing velocify as a function of the terminal falling velocify of a particle. This requires the terminal falling velocify Ut to be expressed in terms of fhe Galileo number. Thus, treating the Stokes, transition and Newton regions in turn ... 

Figure 1.16 Ratio of terminal falling velocity to minimum fluidizing velocity as a function of Galileo number. Reprinted from Davidson, J.F. and Harrison, D., Fluidization, Academic Press, 1971, with permission from Elsevier.

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