Yang Fluidization, Solids Handling, and Processing

as a ratio of the cross sectional average, is roughly invariant over the cross section for these larger beds, the thickness of the wall layer follows from a mass balance. The upflow is proportional to the core area and the downflow to the product of perimeter and layer thickness. Thus the thickness should vary as the ratio of cross sectional area to perimeter, i.e., proportional to the hydrodynamic diameter.

Patience et al. (1992) developed a dimensionless correlation for the mean slip factor between the gas and solid by using solid suspension data from various small laboratory beds. The proposed correlation relates the slip to the Froude number based on the bed diameter. It remains to be seen if the correlation will hold at Froude numbers typical of large beds and if other dimensionless factors are important.

3.4Flow Transition

Hovmand and Davidson (1971) review Stewart’s criterion for the transition from bubbling to slug flow,

and show it gives good agreement with most experiments. Thus, a small diameter laboratory bed may exhibit behavior which is far different than a scaled up pilot plant. There is some data which suggests the bed depth to bed diameter may also influence the bubble to slug transition.

Thiel and Potter (1977) carried out slugging flow experiments in three different diameter beds, the largest being 0.22 m in diameter. They found that the bed aspect ratio influenced the onset of slug formation. They also found that the transition to turbulent flow occurred at much lower superficial gas velocity in the 0.22 m diameter bed than in the 0.1 m bed. With the fluid cracking catalyst, the transition velocity decreased from 20 cm/s to 2.5 cm/s. Yerushalmi and Avidan (1985) assert that in large shallow beds where slugging does not occur, transition from bubbling to turbulent flow should be independent of bed size.

26Fluidization, Solids Handling, and Processing

4.0EXPERIMENTAL MEANS TO ACCOUNT FOR SCALEUP: USE OF SCALE MODELS

Since large pilot plants are costly, it may not be feasible to undertake an extensive scale-up program. Furthermore, large beds are difficult to instrument to observe detailed behavior. Thus, many test results from beds at elevated temperatures and pressure are confined to overall operating characteristics leaving the designers to speculate on the cause of shifts in performance with bed size. Full scale cold models are also expensive and require lengthy construction schedules. As we will see below, the full scale cold test may not correctly simulate the hydrodynamics of the actual process at elevated temperature and pressure. Indeed, a familiar occurrence in atmospheric bubbling fluidized bed combustors is the marked difference in flow behavior between a bed fluidized with cold air and the same bed run at normal operating conditions.

A technique which can assist in the scale-up of commercial plants designs is the use of scale models. A scale model is an experimental model which is smaller than the hot commercial bed but which has identical hydrodynamic behavior. Usually the scale model is fluidized with air at ambient conditions and requires particles of a different size and density than those used in the commercial bed. The scale model relies on the theory of similitude, sometimes through use of Buckingham’s “pi theorem,” to design a model which gives identical hydrodynamic behavior to the commercial bed. Such a method is used in the wind tunnel testing of small model aircraft or in the towing tank studies of naval vessels.

Once a technique has been established to design a model which simulates the hydrodynamics of a hot (possibly pressurized) fluidized bed, then a series of different sized models can be used to determine the influence of bed size on the performance of commercial beds, see Fig. 20. Model A´simulates the behavior of commercial bed A, model B´simulates a larger commercial bed B and so forth. Then by comparing models A´, B´ with C´ we can determine the expected changes in operating characteristics when commercial bed A is replaced by larger beds B and C.

Designing a model fluidized bed which simulates the hydrodynamics of a commercial bed requires accounting for all of the mechanical forces in the system. In some instances, convective heat transfer can also be scaled but, at present, proper scaling relationships for chemical reactions or hydromechanical effects, such as particle attrition or the rate of tube erosion, have not been established.

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exact same equations and boundary conditions as the commercial bed, with the same values of the coefficients, then the model and the commercial bed should have identical behavior. The trick is to nondimensionalize all of the terms of the equations so that the model and the commercial beds have identical nondimensionalized values of the parameters. That is, if the coordinates are nondimensionalized by dividing them by the bed diameter,

and the local gas velocity in the x direction is nondimensionalized by dividing it by the superficial gas velocity u′= uuo , then at the same value of x´ and y´ in a commercial bed and a properly scaled model, u´ will have the same value. Note, this nondimensionalization will extend to time as well. Generally the physical values of x, y, u, and t will be different in the model and the commercial bed, while the dimensionless values x´, y´, u´ and t´ are the same for the two beds.

It is important to include all of the relevant physical effects in the equations to be nondimensionalized. This can be difficult because there isn’t always consensus about which effects are important. Moreover, there is controversy over how to properly represent these effects in equation form. For our purposes the question is: Have all of the important parameters been included in the nondimensional equations? Pragmatically, the success to date of the scaling experiments using the formulation as presented adds confidence to the use of these simplifications which will be employed. Also, a limited number of tests have verified the omission of parameters specifically related to several phenomena.

In some cases, if we are too precise and include effects of marginal significance, the resulting modeling rules may be overly restrictive, limiting the choice of models or requiring the models which are too big. These questions are most expeditiously settled by experiments since an exact theoretical answer is not presently available.

The equations of motion can either be formulated for individual particles and the surrounding fluid, or the fluid and the particulate phases can each be considered a continuum. Both approaches yield identical results, see Glicksman et al. (1994) for a complete derivation. For our purposes, we will base the derivation on the continuum model formulated by Jackson.

4.2Governing Equations

The use of governing equations has the merit of tying specific dimensionless parameters to particular physical phenomena (Glicksman, 1984, 1988). If the proper equations can be written, even if they cannot be solved, they yield considerable insight into the process.

Early derivation of these dimensionless parameters was based on a continuum model (Scharff et al., 1978; Glicksman, 1984). Inclusion of the individual particle approach extends the results to instances where a continuum model may not be applicable.

For simplicity, the fluid will be considered incompressible although the results do not depend on this condition. The continuum equations will be developed similar to those given by Anderson and Jackson (1967). These equations are based on a continuum picture of the suspension of solids in the fluid. The solids and the fluid are considered individual phases, and variables for each phase such as velocity or voidage are average values over regions large compared to a single particle but small compared to the scale of the macroscopic bed behavior. The derivation follows that given by Jackson (1971).

For the fluid, conservation of mass is

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At the bottom

at y = 0

where Gs is the average solids feed rate per unit area from outside the bed through the bottom. This form holds when the solids feed is uniformly distributed over the bottom. If solids feed takes place along the side wall, an expression similar to Eq. (13) must be used in place of Eq. (12).

For gas velocity at the walls,

perforated plate and uo is the superficial gas velocity. For bubble caps or other gas inlet devices, boundary conditions such as Eq. (15) apply over the gas inlet plane of the device. Boundary conditions, Eqs. (12) and (14), also apply to the surfaces of any internal elements in the bed, for example: heat transfer tubes.

Nondimensionalizing the variables as,

where L is a typical bed dimension, e.g., the bed height or diameter. Note that u´, υ´ as well as ε are dependent variables of x ′, y ′, and t ′ which are determined by the boundary conditions and the independent dimensionless parameters of the governing equations.

Applying these definitions to the governing equations and rearranging them so the parameters form nondimensional groups, the continuity equations become

The fluid-particle interaction force, omitting the virtual mass term and combining the pressure terms in the equation of motion becomes