Scaling relationship

Several attempts have been made to determine proper scaling relationships for Group B fluidized beds. These early attempts, and a rigorous... 

For averages of n obsei vations, the corresponding relationship for the Z-scale relationship is... 

In order to get a better notion of the scaling relationship between gyration radius and average chain length for different density regimes, it is convenient... 

The inset shows the diffusion coefficient D vs u scaling relationship [66] with an exponent 0.31. 

Positive Displacement Motor Performance. Using the positive displacement motor performance data in Table 4-114 and the scaling relationships in Equations 4-168 through 4-170, the performance graph for the positive displacement motor operating with a circulation flowrate of 348 gal/min can be prepared. This is given in Figure 4-213. 

Figure 7-10 shows durometer scale relationships and hardness ranges. The letter designations refer to the Shore hardness test (Chapter 5, MECHANICAL PROPERTY, Hardness). 

Solution Table 5.1 provides the scaling relationships, throughput and volume scaling factor is A = 128. 

Since the translation variable, u, is also discretized (dyadically, or uniformly at each scale), relationship (20) will hold only if the signal is translated in time by a period that is an integer multiple of the discretiza-... 

Figure 9 Scaling relationship between root-mean-square radius and molecular weight for polystyrene using GPC/MALLS. (From Wyatt, P. Hicks, D. L., Jackson, C., and Wyatt, G. K., Am. Lab., 20, 108,1988. With permission.)...

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. 

There are several approaches to developing the correct scaling relationships. Probably the most straightforward is the nondimensional-ization of the governing equations. If we can write the proper equations governing the fluid and particle dynamic behavior, we can develop the proper scaling relationships even if we can t solve the equations (at present we can t). In essence, if a model is designed which follows the... 

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. 

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

Figure 24. Error in drag coefficient for simplified scaling relationships, ua/umj = 10. (From Glicksman et al., 1993b.)...

The simplified scaling relationships, Eq. (53), offer some flexibility in the model design since fewer parameters must be matched than with the full set of scaling relationships. When the fluidizing gas, the pressure and temperature of the scale model are chosen, the gas density and viscosity for the scale model are set. The model must still be geometrically similar to the commercial bed. There is still one free parameter. Generally this will be the linear scale of the model. For the simplified scaling relationships, the gas-to-solid density ratio must be maintained constant... 

Note that in simplified case, the velocity scaling is not uniquely tied to just the gas properties as it is in the full scaling relationship. With uQ and ps set, the solids recycle rate can be determined by... 

Since the gas-to-solid density ratio of the model and the commercial beds must be the same to satisfy the simplified scaling relationships, Eq. (87) combined with Eqs. (81) and (78) becomes,... 

Table 4. Scale Models of Atmospheric Commercial Hot Bed Using the Simplified Scaling Relationship...

Table 6 presents the range of scaling parameters for tests undertaken by different investigators to verify the scaling relationships for a bubbling bed. 

Most early experiments devoted to verifying the scaling relationships have dealt with the full set of scaling relationships. Several more recent experiments have dealt with a reduced set of dimensionless parameters. In some experiments, additional scaling parameters were unintentionally matched. 

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