Yang Fluidization, Solids Handling, and Processing

at y = 0

u ′ = i/ above distribution holes

Eq. (27)

u ′ = 0 elsewhere

small compared to sonic velocity or the absolute pressure does not change enough to influence the thermodynamic properties of the fluid; it will be ignored in this development. Note that the fluid pressure level still influences the fluid density.

From the continuity and momentum equations for the fluid and solid phases along with the boundary conditions, the following groups of independent dimensionless parameters are found to control the hydrodynamics, noting our assumption that the particle-particle forces are only dependent on hydrodynamic parameters,

as well as bed geometry.

The dimensionless fluid pressure is not included since it is a dependent parameter.

4.3Fluid-Solid Forces

The drag coefficient β can be expressed in several different limiting forms depending on the flow conditions. At low voidages typical

5.0SIMPLIFIED SCALING RELATIONSHIPS

In later sections, the use of the scaling relationships to design small scale models will be illustrated. For scaling to hold, all of the dimensionless parameters given in Eqs. (36), (37) or (39) must be identical in the scale model and the commercial bed under study. If the small scale model is fluidized with air at ambient conditions, then the fluid density and viscosity are fixed and it will be shown there is only one unique modeling condition which will allow complete similarity. In some cases this requires a model which is too large and unwieldy to simulate a large commercial bed.

To allow more flexibility, we will explore means of simplifying the scaling relationships by reducing the number of parameters which must be maintained constant. In most situations, one would expect that not all of the parameters are of first order importance. By reducing the number of parameters which must be maintained in the model, it may be possible to model larger commercial beds with small scale models. We will look at simplifications of the interparticle drag at the extreme of small and large Reynolds numbers based on particle diameter. This will span the range from low velocity FCC bubbling beds to high velocity fast beds or pneumatic transport using large particles. If the same simplification can be shown to hold in both of these limits, it is reasonable to consider application of the simplification over the entire range of conditions.

5.1Low Reynolds Number

At low particle Reynolds numbers for a bubbling bed, the Ergun expression can be simplified using only the first term in Eq. (31).

Thus,

40 Fluidization, Solids Handling, and Processing

For gas fluidized beds where ρs - ρf can be replaced by ρs,

Substituting Eq. (42) into Eq. (40),

Thus, in the low particle Reynolds number limit, maintaining uo /umf , εmf , and Fr identical between two fluidized bed guarantees that β L /ρsuo is also identical. Although φ and dp are eliminated between Eqs. (40) and (42), in general, particle sphericity and dimensionless size distribution should be held constant in the scaling since they influence εmf . The use of φ and a mean diameter in the Ergun expression only approximates the effects of these parameters. Note if the two models display identical dynamic characteristics, then ε is a dependent variable whose distribution throughout the bed should be identical for both fluidized beds. In this limit, the governing parameters given in Eq. (36) can be expressed as

where εmf will be a function of particle sphericity and size distribution.

This will be referred to as the simplified scaling relationship. At low Reynolds numbers, this still includes the gas-to-particle density ratio.

5.2High Reynolds Numbers

Consider the limit of high particle Reynolds numbers where the inertial term in the Ergun equations dominates.

In this limit,

where u´ = u/uo and v´ = v/uo and u´- v´ is a dimensionless slip velocity. The minimum fluidization velocity can be expressed as,

rearranging and using ρs in place of ρs - ρf ,

Substituting this into Eq. (46) and multiplying by Fr,

Eq. (49)

At large particle Reynolds numbers, just as at low Reynolds numbers, the dimensionless drag, βL/ρsuo, is identical when uo /umf , εmf

42 Fluidization, Solids Handling, and Processing

and Fr are identical. The variables ε, u´ and v´ are dependent dimensionless variables which are identical for two similar fluidized beds. In this limit the same set of governing dimensionless parameters applies as in the low Reynolds number limit, given by Eq. (45).

5.3Low Slip Velocity

Finally, consider the case when the magnitude of the slip velocity between the particles and the gas is close to umf /ε everywhere in the fluidized bed. With the vertical pressure drop equal to the particle weight, the following holds for any value of the particle Reynolds number,

Again, when uo /umf and Fr are identical for two beds and the slip velocity is close to umf /ε, the dimensionless drag coefficient is also identical for two beds.

For all three limiting cases identified above, similitude can be obtained by maintaining constant values for the dimensionless parameters,

5.4General Case

Since the same simplified set of dimensionless parameters holds exactly at both high and low Reynolds numbers, it is reasonable to expect that it will hold, at least approximately, over the entire range of conditions for which the drag coefficient can be determined by the Ergun equation or an equation of similar form.

The validity of the simplified parameters can be checked numerically for the intermediate range of values.

From Eq. (31),

Eq. (54)

where u´- v´ can also be represented as uslip /uo. Using the Ergun equation to determine umf ,

Eq. (55)

where Re = ρf uo dp /μ .

44 Fluidization, Solids Handling, and Processing

It is easy to verify the three limits defined previously by use of Eq. (56).

For the more general case, Fig. 21 shows the value of β given by Eq. (56) relative to β at low Re over a range of conditions when uo /umf is 10 and 3, respectively, and Fr and φs remain constant. When uo /umf and the slip velocity are high, there is a larger variation of dimensionless drag coefficient with Reynolds number. Note that β does not vary with particle Reynolds number when the Reynolds number remains above about 103 or below about 10. Figure 22 illustrates the results when uo /umf is 1000, a condition approached with very fine particle bubbling beds or circulating beds. In the latter instance, the use of the Ergun relationship is questionable except for the dense lower part of the bed.

5.5Range of Validity of Simplified Scaling

To determine the validity of the simplified scaling laws over a wide range of conditions the simplified scaling laws have been used (Eq. 53) to design hypothetical models whose linear dimensions are 1/4 and 1/ 16, respectively, of the linear dimensions of a model designed using the full set of scaling laws, Eq. (37). To determine the validity of the smaller, simplified models, the dimensionless drag coefficient β L/ρsuo will be compared between the simplified models and the model using the full set of scaling laws. Figure 23 shows a comparison of the exact model and the simplified models for a pressurized fluidized combustor. Using the full set of scaling laws, the exact model, fluidized by ambient air, is approximately the same size as the combustor. The simplified models are reduced in size by their respective assumed length scale. The other parameters of the simplified model are then calculated to match the simplified parameters. For example, when the length scale is reduced to 1/4 that of the exact model, the velocity is reduced by 1/2 to keep the Froude number constant. The particle diameter is then reduced appropriately to keep the ratio of uo/umf constant. These calculations were carried out over a range of particle Reynolds numbers, RepE, based on the full scaling law, or exact, model. In the simplified scaling relationships, the Reynolds number is not maintained constant. The concern is how much the drag coefficient is impacted by the shift in Reynolds number. It was found that the particle Reynolds number for the 1/4 scale simplified model remained roughly equal to 0.34 RepE over a wide range of values for RepE, whereas the particle Reynolds number for the 1/16 scale model was roughly 0.12 RepE.