Hydrodynamic scaling

The scaling dependence of the diffusion coefficient on N and Cobs Iso poses a number of questions. While the original scaling predictions, based on reptation dynamics [26,38], oc N, have been verified by some measurements [91,98], significant discrepancies have been reported too [95,96]. Attempts to interpret existing data in terms of alternative models, e.g., by the so-called hydrodynamic scaling model [96], fail to describe observations [100,101]. 

D. Maynes, J. Klewicki, P. McMurty, H. Robey. Hydrodynamic scalings in the rapid growth of crystals from solution. J Cryst Growth 178 545, 1997. 

Phillies, GDJ, The Hydrodynamic Scaling Model for Polymer Self-Diffusion, Journal of Physical Chemistry 93, 5029, 1989. 

Farrell, P. A., Hydrodynamic Scaling and Solids Mixing in Pressurized Fluidized Bed Combustors, Ph.D. Thesis, Massachusetts Institute of Technology (1996)... 

Glicksman, L. R., and Farrell, P., Verification of Simplified Hydrodynamic Scaling Laws for Pressurized Fluidized Beds Part I Bubbling Fluidized... 

Glicksman, L. R., Hyre, M., Torpey, M., and Wheeldon, J., Verification of Simplified Hydrodynamic Scaling Laws for Pressurized Fluidized Beds Part II Circulating Fluidized Beds, Proc. 13th Int. Conf. for Fluidized Bed Combustion, ip. 991 (1995)... 

Phillies GDJ (1995) Hydrodynamic scaling of viscosity and viscoelasticity of polymer solutions, including chain architecture and solvent quality effects. Macromolecules 28(24) 8198-8208... 

Phillies, G., The hydrodynamic scaling model for polymer self-diffusion. Journal of Physical Chemistry, 1989, 93, 5029-5039. 

Deducing gas phase hydrodynamics based on radial and axial concentration measurements is the ideal manner in which to scale-up. However, this approach is often prohibitive in terms of cost and time investment so we rely on cold flow experiments for hydrodynamic scaling. 

Figure 12.1 In hydrodynamic scale-up the velocity gradients are similar, but the absolute velocities are not.

Although the theories above lead to a very favorable throughput in the production machine, one of the major problems in extrusion with large machinery has not yet been dealt with. Geometrical and hydrodynamic scale-up do not take the temperature distribution and the heating by viscous dissipation or exothermic reactions in the extruder into account. 

Let us now employ the mass conservation law written in an hydrodynamic scale ... 

The second way is expressing Mi in terms of tracer diffusivities D in a two-phase alloy [38]. Assuming the Kirkendall effect is suppressed in the hydrodynamics scale, we finally have... 

Huge problems arise due to lack of scale separation. The assumption of fast local equilibrium cannot be employed a priori except for the nearly elastic system with slow change of the external parameters. As a result, the hydro-dynamic description, which demands that the mean free time is far less than the hydrodynamic scale, cannot be satisfied for granular materials, as indicated in Du et al (1995) by showing the breakdown of hydrodynamics in a one-dimensional system of inelastic particles. 

Fitzgerald and Crane (1980) were among of the first to evaluate the full set of hydrodynamic scaling parameters. They compared the hydrodynamics of two scaled beds using pressure fluctuation measurements and movies. In one bed cork particles were fluidized with air the other bed used sand fluidized with pressurized refrigerant 12 vapor. Movies showed qualitative agreement between bubble growth and the solids flow in the beds. 

Farrell PA. Hydrodynamic scaling and solids mixing in pressurized fluidized bed combustors. PhD dissertation, Massachusetts Institute of Technology, Cambridge, MA, 1996. 

Glicksman LR, H5re M, Torpey M, Wheeldon J. Verification of simplified hydrodynamic scaling laws for pressurized fluidized beds Part II. Circulating fluidized beds. Proc. 13th Int. Conference for Fluidized Bed Combustion, 1995, p 991. 

There have been a number of efforts to introduce a sufficient set of dimensionless groups which, when matched and coupled with geometric similarity, assure dynamic similarity between small- and large-scale units. This work is well summarized by van der Meer et al. (1999). For full hydrodynamic scaling, they recommend that at least five dimensionless groups be matched ... 

On the other hand, the existence of a concentration-independent crossover probe diameter d (R, R is consistent with polymer solution models based on an assumed dominance of hydt ynatnic interactions in nondilute solution. Models such as the hydrodynamic scaling model (6.7) identify the chain radius as the primary solution length scale at all concentrations at which the model applies. With this identification, a crossover from small-probe to large-probe behavior, perhaps correlated with differential ability to interact with internal chain modes, would at all concentrations occur over the same range of d/R, or d/R, precisely as observed experimentally. 

G. D. J. Phillies. Derivation of the universal scaling equation of the hydrodynamic scaling model via renormalization group analysis. Macromolecules, 31 (1998), 2317-2327. 

Figure 8.36 Scaling pre factora as a function of M using results from Refs. (O) (4) for polystyrene in CCI4, ( ) (16), ( ) (25,24), (A) (9), (A) (21,23) with linear chains, (+) (21,22) with/ = 3, (X) (21) with / = 8, (El) (23) with /=18, (1), ( ) (2), (V) (4) for polymers in C6D6, ( ) (10), (0) (27), ( ) (26), ( ) (22) for linear polybutadiene, ( ) (22) for / = 3 polybutadiene, (>) (2) for PEO in water, and (<) (3) for xanthan in water. Dashed line indicates best-fit line with a jj O.98 Solid line is the no-free-parameter prediction of a from the hydrodynamic scaling model, Chapter 17. Other details as in Figure 8.34.

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