Hydrodynamic scaling parameters

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. 

Fitzgerald et al. (1984) measured pressure fluctuations in an atmospheric fluidized bed combustor and a quarter-scale cold model. The full set of scaling parameters was matched between the beds. The autocorrelation function of the pressure fluctuations was similar for the two beds but not within the 95% confidence levels they had anticipated. The amplitude of the autocorrelation function for the hot combustor was significantly lower than that for the cold model. Also, the experimentally determined time-scaling factor differed from the theoretical value by 24%. They suggested that the differences could be due to electrostatic effects. Particle sphericity and size distribution were not discussed failure to match these could also have influenced the hydrodynamic similarity of the two beds. Bed pressure fluctuations were measured using a single pressure point which, as discussed previously, may not accurately represent the local hydrodynamics within the bed. Similar results were... 

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. 

Farrell (1996) experimentally evaluated the importance of the solid-to-gas density ratio (Ps/pf) for scaling the hydrodynamics of bubbling and slugging fluidized beds. Two bed materials, polyethylene plastic (pg = 918kg/m ) and a dolomite/limestone sorbent mixture (ps = 2670 kg/m ), were used to create a mismatch in the density ratio. The size of the particles was chosen so that the remaining simplified scaling parameters were matched. The internal angle of friction was similar between the two materials. 

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.

Here the scaling exponent v and scaling prefactor a are independent of polymer concentration, but may depend on polymer molecular weight. Phillies hydrodynamic scaling model predicts not only the molecular weight dependences of v and a but also approximate numerical values for both parameters(14). 

The Navier-Stokes description of a fluid is more coarse-grained than the original LB equation, and to connect the microscopic scales with the hydrodynamic scales we follow a standard asymptotic analysis [103]. We first introduce a dimensionless scaling parameter e 1 and write... 

The function / incorporates the screening effect of the surfactant, and is the surfactant density. The exponent x can be derived from the observation that the total interface area at late times should be proportional to p. In two dimensions, this implies R t) oc 1/ps and hence x = /n. The scaling form (20) was found to describe consistently data from Langevin simulations of systems with conserved order parameter (with n = 1/3) [217], systems which evolve according to hydrodynamic equations (with n = 1/2) [218], and also data from molecular dynamics of a microscopic off-lattice model (with n= 1/2) [155]. The data collapse has not been quite as good in Langevin simulations which include thermal noise [218]. 

The methodology discussed previously can be applied to the study of colloidal suspensions where a number of different molecular forces and hydrodynamic effects come into play to determine the dynamics. As an illustration, we briefly describe one example of an MPC simulation of a colloidal suspension of claylike particles where comparisons between simulation and experiment have been made [42, 60]. Experiments were carried out on a suspension of AI2O3 particles. For this system electrostatic repulsive and van der Waals attractive forces are important, as are lubrication and contact forces. All of these forces were included in the simulations. A mapping of the MPC simulation parameters onto the space and time scales of the real system is given in Hecht et al. [42], The calculations were carried out with an imposed shear field. 

Scaling by use of dimensionless numbers only is limited in two-phase flow to simple and isolated problems, where the physical phenomenon is a unique function of a few parameters. If there is a reaction between two or more physical occurrences, dimensionless scaling numbers can mainly serve for selecting the hydrodynamic and thermodynamic conditions of the modelling tests. In... 

Another approach to scale-up is the use of simplified models with key parameters or lumped coefficients found by experiments in large beds. For example, May (1959) used a large scale cold reactor model during the scale-up of the fluid hydroforming process. When using the large cold models, one must be sure that the cold model properly simulates the hydrodynamics of the real process which operates at elevated pressure and temperature. 

Ishii and Murakami (1991) evaluated the CFB scaling relationships of Horio et al. (1989) using two cold CFB models. Solids flux, pressure drop, and optical probe measurements were used to measure a large number of hydrodynamic parameters to serve as the basis for the comparison. Fair to good similarity was obtained between the beds. Dependent hydrodynamic parameters such as the pressure drop and pressure fluctuation characteristics, cluster length and voidage, and the core diameter were compared between the two beds. The gas-to-solid density ratio was not varied between the beds. As seen in Table 7, the dimensionless solids flux decreased as the superficial velocity was increased because the solids flux was held constant. 

The average DNA helix diameter used in modeling applications such as the ones described here includes the diameter of the atomic-scale B-DNA structure and— approximately—the thickness of the hydration shell and ion layer closest to the double helix [18]. Both for the calculation of the electrostatic potential and the hydrodynamic properties of DNA (i.e., the friction coefficient of the helix for viscous drag) a helix diameter of 2.4 nm describes the chain best [19-22]. The choice of this parameter was supported by the results of chain knotting [23] or catenation [24], as well as light scattering [25] and neutron scattering [26] experiments. 

Dobetti and Pantaleo (38) investigated the influence of hydrodynamic parameters per se on the efficiency of a coacervation process for microcapsule formation. They based their work on that of Armenante and Kirwan (39) who described the size of the smallest eddies or vortices generated in a turbulent regime on a microscopic scale in the vicinity of the agitation source, i.e., microeddies, as... 

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