## Q O Superficial Velocity

Figure 18.3 Idealized efleet of superficial velocity on the pressure drop and bed height of a fluidized bed.

height to increase. Any particle that rises above the top of the bed due to transient nonuniformities in flow will encounter a local upward velocity that is less than the particle's terminal settling velocity and the particle will fall back into the bed. Ultimately, a point (B) will be reached at which the local fluid velocity around the particles is equal to their terminal settling velocity. If the superficial velocity is increased beyond that point the upward drag forces on the particles will exceed the downward gravitational forces and the particles will be carried away and the bed will cease to exist. It is then in continuous fluidization.

Consideration of Figure 18.3 suggests that two velocities are critical to defining the operating range for a fluidized bed, those associated with points A and B. The velocity associated with point A is called the minimum fluidization velocity. The formula for its computation can be derived by equating the pressure drop as given by the Ergun equation to the weight of the bed per unit area of cross-section, allowing for the buoyant force of the displaced water. This results in a quadratic equation for the minimum fluidization velocity, vml:

in which dp is the diameter of the particle, pn is its density, pw is the density of water, p.„ is the viscosity of water, g is gravitational acceleration, and eM is the minimum porosity at incipient fluidization, i.e, point A. For roughly spherical particles, eM is generally between 0.40 and 0.45, decreasing slightly with increasing particle size. The velocity associated with point B is the terminal settling velocity of the carrier particles, v,. If it is exceeded the particles are carried away in continuous fluidization and the bed is destroyed. The equation for its computation is derived by equating the drag force on the particle to the gravitational force minus the buoyant force. The result is:

in which C0 is the drag coefficient. The value of Cn depends on the Reynolds number and correlations are available to relate the two for spherical particles as well as for other shapes. The appropriate velocity to use in computation of the Reynolds number is the terminal settling velocity of the particle. The result is usually called the terminal Reynolds number, Re,. Because the terminal settling velocity depends on the Reynolds number (through the drag coefficient) and the Reynolds number depends on the velocity, an iterative procedure may be required to compute the terminal settling velocity, depending on the nature of the CD vs. Re, relationship. Techniques are available for defining ranges of Reynolds numbers over which direct solutions may be possible and the reader should consult other sources to learn more about them.*

The height that a fluidized bed of clean carrier particles attains (H]tp) depends directly on the porosity that results from the applied superficial velocity. This follows from the fact that the mass of particles in the bed is constant. Therefore:

where HKp is a reference bed height and eR is the porosity associated with it. Some use the minimum bed height immediately prior to fluidization as HRp, in which case, ek will be em.* Others21avoid the need to know eM by using as HRp the height that would be occupied by the carrier particles if they formed a solid block with mass equal to the total mass of carrier particles present, in which case er would be zero and HKp would be given by:

PpA, where Mp is the mass of carrier particles and is the cross-sectional area of the FBBR. Substitution of Eq. 18.4 into Eq. 18.3 gives:

which can be used to calculate the height of clean carrier particles in a fluidized bed. However, regardless of which definition of HRp is used, prediction of the bed height associated with a given superficial velocity requires prediction of the porosity, e. This can be done by using the Richardson-Zaki equation: