## Terminal Reynolds Number

Figure 18.5 Effect of terminal Reynolds number, Re„ on the drag coefficient, C,„ of bio-particles. (From K. S. Ro and J. B. Neethling, Terminal settling characteristics of bioparticles. Research Journal, Water Pollution Control Federation 62:901-906, 1990. Copyright <' Water Environment Federation; reprinted with permission.)

Figure 18.5"J shows three relationships and compares them to the relationship of Schiller et al. (referenced in 24) for clean spherical particles. The equations are given in Table 18.1. Two things are evident from the figure. First, the growth of a biofilm increases the drag coefficient relative to that of a clean particle with equivalent terminal Reynolds number, i.e., equivalent diameter and density. Second, the rela- Table 18.1 Equations Depicting the Relationships Between the Drag Coefficient and the Terminal Settling Velocity of Bioparticles Shown in Figure 18.5

Equation Source

C|, = 17.1 Re, " ' Hermanowicz and Ganczarczyk "

C„ = 24 Re, + 21.55 Re, "ils Ro and Neethling''

tionships found by the three studies on biofilms are all different, suggesting that the influence of biofilm growth on C„ may be case specific.

Figure 18.6~J shows the effect of biofilm growth on the terminal settling velocity of bioparticles in which sand (p,, = 2.65 g/crrT) with a diameter of 0.5 mm serves as the carrier particle. The values were calculated with Eq. 18.2 using the bioparticle density, ph, from Eq. 18.8 in place of pr and the bioparticle diameter, d,„ Smooth, equivalent density spheres

### Spherical bioparticles

Figure 18.6 Effect of relative biofilm thickness, d^'d,,, on the terminal settling velocity of spherical and nonspherical bioparticles with sand carrier particles with a diameter, d,„ of 0.5 mm. For comparison, the terminal settling velocity of smooth, equivalent density spheres is also shown. (From K. S. Ro and J. B. Neethling, Terminal settling characteristics of bioparticles. Research .Journal, Water Pollution Control Federation 62:901-906, 1990. Copyright © Water Environment Federation; reprinted with permission.)

Smooth, equivalent density spheres

Spherical bioparticles

Nonspherical bioparticles (psi and phi = 1.1)

Relative Biofilm Thickness (db/dp)

Figure 18.6 Effect of relative biofilm thickness, d^'d,,, on the terminal settling velocity of spherical and nonspherical bioparticles with sand carrier particles with a diameter, d,„ of 0.5 mm. For comparison, the terminal settling velocity of smooth, equivalent density spheres is also shown. (From K. S. Ro and J. B. Neethling, Terminal settling characteristics of bioparticles. Research .Journal, Water Pollution Control Federation 62:901-906, 1990. Copyright © Water Environment Federation; reprinted with permission.)

from Eq. 18.7 in place of d,,. The value of C,, was computed from the correlation of Ro and Neethling:J shown in Figure 18.5. Three important points are evident in the figure. First, the terminal settling velocity of the bioparticles decreases as the biofilm thickness increases. Since terminal settling velocity is directly proportional to the diameter of a particle (see Eq. 18.2), the decrease in terminal settling velocity associated with an increase in biofilm thickness is due to the decrease in the effective density of the bioparticle (see Eq. 18.8). Second, as the biofilm thickness increases, a point is eventually reached at which further increases have little effect. In that region the effects of increases in diameter are approximately equal to the effects of decreases in density. Third, the settling velocity of a bioparticle is always lower than that of a smooth sphere of equivalent diameter and density. This is due to the effect of the biofilm on the drag coefficient. The latter point is true for a wide range of carrier particle sizes and densities, as well as for a broad range of biofilm thicknesses, with the effect that the settling velocity of a bioparticle is always between 55 and 60% of the velocity of an equivalent density smooth sphere of the same diameter. 4

The effects of particle density are shown in Figure 18.7 " for a case in which the growth of the biofilm has no effect on the relationship between the drag coefficient and the terminal Reynolds number. In other words, it assumes that clean carrier particles have the same surface characteristics as those with biofilm. There it can be seen that biofilm growth can increase the settling velocity of carrier particles of low density. In fact, the counteracting effects of the changes in density and diameter can make the settling velocity of bioparticles containing low density carrier particles change in complex ways as they grow larger, particularly when the effects on the drag coefficient are also considered. This can have a significant effect on the migration of bioparticles in FBBRs.

Bed Porosity and Expansion. Because growth of a biofilm changes the terminal settling velocity .of a particle, it also changes its fluidization properties. One effect is on the porosity associated with a given superficial velocity. According to the Richardson-Zaki equation (Eq. 18.6) if the superficial velocity is held constant and the terminal settling velocity of a particle is changed, the porosity of the bed will change. This, is turn will change the height of the fluidized bed, as indicated by Eq. 18.3. Another effect is on the reference bed height. Particles with a larger diameter occupy more space. Thus, the reference bed height will be larger, which will cause the expanded bed height to increase as well. Because the volume of a sphere is proportional to its diameter cubed, the value of the reference bed height for bioparticles, H ki,, can be related to the mass of carrier particles present by:

This equation can be substituted into Eq. 18.3 to give the height of a fluidized bed containing bioparticles:

The Richardson-Zaki equation (Eq. 18.6) has been shown to be applicable to bioparticles, although the correlation between the coefficient n and the Reynolds or Galileo number is different from that for clean particles.1'" " One that works well