Free fall velocity

The particle size deterrnined by sedimentation techniques is an equivalent spherical diameter, also known as the equivalent settling diameter, defined as the diameter of a sphere of the same density as the irregularly shaped particle that exhibits an identical free-fall velocity. Thus it is an appropriate diameter upon which to base particle behavior in other fluid-flow situations. Variations in the particle size distribution can occur for nonspherical particles (43,44). The upper size limit for sedimentation methods is estabHshed by the value of the particle Reynolds number, given by equation 11 ... 

Economic considerations dictate that a commercial fluid bed operates at as high a gas velocity as practicable. A frequent limiting factor is entrainment from the bed which is a very strong ftmction of gas velocity. A well defined fluid bed can be maintained even at gas velocities well in excess of the free fall velocity of the biggest particles... 

We Starr by considering the free-falling velocity of a single particle of diameter (if. When the particle reaches the free falling-velocity, w, the gravitational force and the drag force are in equilibrium, after which the falling velocity is constant. 

Here we consider the problem from another point of view, namely how a set of particles of different sizes fall in a gas. In other words, we wish to find a simple way to calculate the free-falling velocity of a group of particles when we know the distribution of the size of the particles. But before we can do this, some basic widely-used formulas for calculating the free-falling velocity of a separate single particle are introduced. 

Substituting this into Eq. (14.19) and then combining this with the definition of Re, Eq. (14.20), and the force equilibrium condition, Eq. (14.22), we obtain the following equation for the free-falling velocity ... 

TABLE 14.1 The Free-Falling Velocity of a Spherical Particle in Air (20 °C, 1.023 bar) based on Stokes s Theory... 

Beard and Pruppacher have examined the free-falling velocity for small water droplets in saturated air. These results could be correlated as follows... 

Combining this with Eqs. (14.19) and (14.22), the corresponding free-falling velocity w q can be determined. 

In the pneumatic conveying process the flow around the particle is not uni form, the particle is not in steady-state motion, and the flow contains turbulence which is not merely generated by the particles. Thus the use of Eqs. (14.23)-(14.29) is of course rather restricted. Despite these limitations we will now esti mate the free-falling velocity of a set of different-sized particles based on the assumption that we know the free-falling velocity of each single particle. 

Particles of different sizes fall at different velocities. When a set of different-sized particles falls in a group, the particles collide with each other and the faster ones tend to accelerate the slower ones. In all collisions the linear momentum is conserved, so that if all particles collide with each other sufficiently many times, the set of particles will achieve one mean free-falling velocity. Thus the mean free-falling velocity of the set of particles can be defined by... 

All other cases are between the extreme limits of Stokes s and Newton s formulas. So we may say, that modeling the free-falling velocity of any single particle by the formula (14.49), the exponent n varies in the region 0.5 s n < 2. In the following we shall assume that k and n are fixed, which means that we consider a certain size-class of particles. 

The corresponding diameter of a single particle with free-falling velocity m/jq is, according to Eq. (14.49),... 

This gives us a mean particle diameter, that has the same free-falling velocity as the set of particles of different sizes. Taking a logarithm of Eq. (14.65), we get... 

Comparing this result with Eq. (14.48) and Fig. 14.5, we see that the mean size of the particle in the sense of linear momentum, and therefore also in the sense of free-falling velocity, is always greater than the mean mass size... 

FIGURE 14.6 Log-normal probability size distributions, illustrating geometrical transposition between number, mass, and linear momentum curves and the mean size particle d,, which can be used in estimating the free-falling velocity of the particle group. 

In the previous section we determined the equivalent particle diameter of a set of particles of different sizes, with the aid of which we can treat the mixture as composed of one size of particles, namely The mean free-falling velocity of the mixture is the same. 

So far we have considered the free-falling velocity in air which is at rest. But the falling phenomenon in a tube differs from this because the falling of solid particles itself causes an airflow upwards (see Fig. 14.7). The modeling idea of Fig. 14.7 is from Weber. The air volume replaced by the falling parti cle flows upward and this airflow rate is... 

In the coordinate system moving upward with velocity the particle is falling as if the air around it were at rest. In other words, the free-falling veloc-itv is the sum of the two velocities... 

Equation (14.30), whose analytical form was Eqs. (14.64) and (14.65), can be used for determining the free-falling velocity w q in the case that the particles behave like separate particles but due to the great number of random collisions have one free-falling velocity w q. Then Eq. (14.71) gives the correction to calculate the falling velocity in tube flow. 

In the following we will give some empirical formulas for the correlation between and w q, where w q is to be understood as a proper mean free-falling velocity of the mixture, but is not necessarily given by the analysis based on Eq. (14.30). 

Elutriation differs from sedimentation in that fluid moves vertically upwards and thereby carries with it all particles whose settling velocity by gravity is less than the fluid velocity. In practice, complications are introduced by such factors as the non-uniformity of the fluid velocity across a section of an elutriating tube, the influence of the walls of the tube, and the effect of eddies in the flow. In consequence, any assumption that the separated particle size corresponds to the mean velocity of fluid flow is only approximately true it also requires an infinite time to effect complete separation. This method is predicated on the assumption that Stokes law relating the free-falling velocity of a spherical particle to its density and diameter, and to the density and viscosity of the medium is valid... 

In this method, the free-falling velocity of particles of selected size, in still air, is counterbalanced by an upward, uniform flow of air or gas at the same (free-falling) velocity. Particles... 

For the transport of a dilute suspension of solids, uR will approximate to the free-falling velocity uq of the particles in the liquid. For concentrated suspensions, a correction must be applied to take account of the effect of neighbouring particles. This subject is considered in detail in Volume 2 (Chapter 5) from which it will be seen that the simplest form of... 

By definition, the terminal velocity of a particle (ut) is the superficial gas velocity which suspends an isolated particle without translational motion—i.e., the terminal free fall velocity for that particle. From force balance on the particle, the terminal velocity for an approximately spherical particle can be shown to be... 

A glass sphere, of diameter 6 mm and density 2600 kg/m3, falls through a layer of oil of density 900 kg/m3 into water. If the oil layer is sufficiently deep for the particle to have reached its free falling velocity in the oil, how far will it have penetrated into the water before its velocity is only 1 per cent above its free falling velocity in water It may be assumed that the force on the particle is given by Newton s law and that the particle drag coefficient R /pu2 = 0.22. 

A binary suspension consists of equal masses of spherical particles of the same shape and density whose free falling velocities in the liquid are 1 mm/s and 2 mm/s, respectively. The system is initially well mixed and the total volumetric concentration of solids is 0.2. As sedimentation proceeds, a sharp interface forms between the clear liquid and suspension consisting only of small particles, and a second interface separates the suspension of fines from the mixed suspension. Using a suitable model for the behaviour of the system, estimate the falling rates of the two interfaces. It may be assumed that the sedimentation velocity uc in a concentrated suspension of voidage e is related to the free falling velocity u0 of the particles by ... 

For particles of glass ballotini with free falling velocities of 10 and 20 mm/s the index n has a value of 2.39. If a mixture of equal volumes of the two particles is fluidised, what is the relation between the voidage and fluid velocity if it is assumed that complete... 

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