Dimensionless number Archimedes

The real coating process in the studied Wurster coater apparatus with the bed mass of 3 kg contains about 21.8x10 particles with the size of 550 im. Unfortunately, the numerical effort for the calculation of the DPM model increases with increasing the number of simulated particles. The DPM model is unable to represent this number of particles, at least with the actually available computing power. However, the number of particles can be reduced by conservation of the particle and fluid dynamics in the simulated apparatus and its real geometry. In this work a scaling approach proposed by Link et al. (2009) and extended by Sutkar et al. (2013) has been used, in which the scahng of the particle size was carried out. Due to the size increase, the adequate properties of sohd and gas phase have been adapted to keep the dimensionless numbers Archimedes At) and Reynolds Re) and the velocities of minimal fluidization and elutriation constant. 

American engineers are probably more familiar with the magnitude of physical entities in U.S. customary units than in SI units. Consequently, errors made in the conversion from one set of units to the other may go undetected. The following six examples will show how to convert the elements in six dimensionless groups. Proper conversions will result in the same numerical value for the dimensionless number. The dimensionless numbers used as examples are the Reynolds, Prandtl, Nusselt, Grashof, Schmidt, and Archimedes numbers. 

It is useful to take similarity principles and dimensionless numbers into consideration when planning experiments. Experiments may involve different levels of velocities and temperature differences. It is important to select values that give a large variation of Archimedes number (12,56) to obtain a high possibility of large physical effects in the measurements. 

Archimedes number A dimensionless number that relates the ratio of buoyancy forces to momentum forces, expressed in many forms depending on the nature of the Reynolds number. 

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. 

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. 

Many models have been proposed to correlate the particle terminal velocity with physical properties, including sphericity. The equation of Haider and Lev-enspiel (1989) depends on the dimensionless number known as the Archimedes number, Nm-... 

This formulation, even if we were able to express function 32, is not easy to use because the fall velocity appears in both dimensionless numbers. Consequently, it is expressed in an equivalent manner as a relation Re = 3 Ar between the Reynolds number and Archimedes number ... 

For completeness, we mention the most common other dimensionless numbers which are frequently encountered in the literature on multiphase flow. These are the Archimedes number, Capillary number, and Ohnesorge number, defined as... 

Determine the settling velocity of spherical quartz particles in water (d = 0.9 mm) using the dimensionless plot of the Lyachshenko and Reynolds numbers versus the Archimedes number in the figure above. The Lyashenko number is the same as the dimensionless settling number. The specific weight of the quartz is 2650 kg/m and the temperature of the water is 20° C. 

This ratio represents an average between similar ratios for the laminar and turbulent regimes. In the most general case, u, = f(D, Pp, p, /r, r, w), and hence we may ignore whether the particle displacement is laminar, turbulent or within the transition regime. This enables us to apply the dimensionless Archimedes number (recall the derivation back in Chapter 5) ... 

Figure 12.41 shows the results of three experiments with a similar Archimedes number and different Reynolds numbers. The figure shows vertical temperature profiles in a room ventilated by displacement ventilation. The dimensionless profiles are similar within the flow rates shown in the figure, although the profile may involve areas with a low turbulence level in the middle of the room. A test of this type could indicate that further experiments can be performed independently of the Reynolds numbers. 

Ar = d3ppfgAp/ju2, Archimedes number, dimensionless At Cross-sectional area based on particle terminal velocity, cm ... 

Either v, or Ea can be calculated from Eq. (9.20), if there is one more relationship between the two terms. In Fig. 9.12, v,. related to the singledrop velocity Vp according to Eq. (9.15) is plotted as a function of the drop holdup for droplet swarms with the Archimedes number as a dimensionless term for the drop diameter for the measured values. It can be seen that the relative velocity constantly decreases, as the holdup of the drops, e, increases and the size of the drops in the swarm decreases. 

ArRe /Ct can be regarded as an expanded Archimedes number and thereby as a dimensionless expression for the tank diameter. The kinematic quantity u drc.min is no longer contained in it.)... 

Mg. 3.6-8 Ratio v /v f- against the particle Archimedes number is plotted against the dimensionless particle diameter... 

Fig. 3.7-5 Dimensionless mean specific power input vs. the Archimedes number...

This method of calculation is satisfactory provided it is known a priori that the Reynolds number is small (< 1). As the unknown velocity appears in both the Reynolds ninnber and the drag coefficient, it is more satisfactory to work in terms of a new dimensionless group, Ar, the so-called Archimedes munber defined by ... 

Equation 2.4 is, actually, the product of the inverse of the cohesion number and another dimensionless group known as the Archimedes number Ar, and thus... 

It must be noted that the Archimedes number Ar T is based on the Sauter diameter referring to Equations (3.81) and (3.69) and y/ denotes the dimensionless sphericity of the particles. 

The same relation is expressed in Equations (3.91) using the dimensionless Reynolds (Re) and Archimedes (Ar) numbers, see Equations (3.87) and (3.81), where is the drag function of a particle or bed depending on the Reynolds number Re and the bed voidage e. For fluid-particle interaction the following states can be distinguished ... 

The above expressions for the terminal velocity of a single particle may be expressed in dimensionless form, thereby introducing another of the dimensionless groups, the Archimedes number Ar, which will subsequently be used in the characterization of the fluidized state. 

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