Hydraulic diameter of the particles

Diameter of spherical particle Average diameter of the particle Sieve passage diameter for 85% of the particles Sieve passage diameter for 50% of the particles Hydraulic diameter of a noncircular flow Inner diameter of pipe... 

Haas, D. B., Gillies, R., Small, M. and Husband, W. H. W. Saskatchewan Research Council publication No. E-835-1-C80 (March 1980). Study of the hydraulic properties of coarse particles of metallurgical coal when transported in slurry form through pipelines of various diameters. 

With turbulent channel flow the shear rate near the wall is even higher than with laminar flow. Thus, for example, (du/dy) ju = 0.0395 Re u/D is vaHd for turbulent pipe flow with a hydraulically smooth wall. The conditions in this case are even less favourable for uniform stress on particles, as the layer flowing near the wall (boundary layer thickness 6), in which a substantial change in velocity occurs, decreases with increasing Reynolds number according to 6/D = 25 Re", and is very small. Considering that the channel has to be large in comparison with the particles D >dp,so that there is no interference with flow, e.g. at Re = 2300 and D = 10 dp the related boundary layer thickness becomes only approx. 29% of the particle diameter. It shows that even at Re = 2300 no defined stress can be exerted and therefore channels are not suitable model reactors. 

Fig. 8. Typical CO concentration and reaction rate profiles in the porous Pt/y-Al203 catalyst reconstructed by particle-packing method. Mean hydraulic diameter of macropores = 300 nm, macroporosity =18.1%. Free space corresponds to macropores, solid gray corresponds to mesoporous y-Al203 with dispersed Pt. Length of the section edge 10 pm. Boundary /.. yco 1%, y0j = 0.5%. (a) T 513 K, (b) T = 533 (Koci et al., 2007a) (see Plate 2 in Color Plate Section at the end of this book).

This correlation gives, for perfectly wettable solids, fairly good estimates of the static holdup for different particle-geometries and sizes. Saez and Carbonnel [26] used the hydraulic diameter, instead of the nominal particle diameter, as the characteristic length in the Eotvos number, to include the influence of the particle geometry on the static hold-up. However, no improvement could be obtained in correlating the data with this new representation. 

Assuming that the pore diameter is characterized by the mean half hydraulic radius of the pore system, further assuming complete wetting and spherical monosized particles, the following equation is obtained ... 

The Taylor-Aris result for the dispersion coefficient (Eq. 4.6.35) has been applied to the empirical correlation of measured and calculated longitudinal dispersion coefficients in flow through packed beds and porous media (see Eidsath et al. 1983). Typically, the velocity in the Peclet number of the Taylor-Aris formula is identified with the superficial velocity, and the capillary diameter with the hydraulic diameter for spherical particles. An alternative velocity suggested by the capillary model is the interstitial velocity, and an alternative length is the square root of the permeability. In an isotropic packing of particles is about one-tenth the particle diameter (Probstein Hicks... 

In a pressure regime high enough, the permeate flux becomes independent of the applied pressure, which is the critical flow of the process. The presence of a layer of particles trapped and compressed on the surface of the membrane leads to the maintenance of a constant pressure drop in the gel layer polarization, and this pressure is the critical pressure of the system. Considering that the thickness of the layer retained on the membrane surface is very small, relative to the diameter of the pore channel, we can neglect its effect in relation to the hydraulic conditions of flow, and thus, the flow on the surface can be given as zero, thus characterizing the critical flow. 

The hydraulic diameter d was defined in Chap. 3. If h denotes the mean distance of the particles in flow direction, then the following equation can be applied for randomly arranged systems of particle volume Fp ... 

Processing of numerical experimental results led to the equation for the minimal diameter of dispersed phase particles undergoing deformation as a result of the continuous hydraulic impact of the medium [66] ... 

Reactant concentration on the outer catalyst surface Reference tracer concentration Diameter of the channel (tube), hydraulic diameter Molecular diffusion coefficient Effective diffusion coefficient Diameter of particle... 

Fluid Kinematics. Water flowing at a steady rate in a constant-diameter pipe has a constant averse velocity. The viscosity of water introduces shear stresses between particles that move at different velocities. The velocity of the particle adjacent to the wall of the pipe is zero. The velocity increases for particles away from the wall, and it reaches its maximum at the center of the pipe for a particular flow rate or pipe discharge. The velocity profile in a pipe has a parabolic shape. Hydraulic engineers use the average velocity of the velocity profile distribution, which is the flow rate over the cross-sectional area of the pipe. 

Spectral methods have been most heavily used in simulations of turbulence. Orszag and Patterson [139] reported the first simulation of turbulent flow. They used spectral methods to simulate homogeneous turbulence in a periodic cubic box. Kim et al. [138] published the first results for fully developed turbulent channel flow. The Reynolds number, based on the hydraulic diameter and the bulk velocity, was roughly 13,660. Their results were generally in good agreement with experimental results obtained for wall-bounded turbulent shear flows. This approach has been widely used to track particles or drops in recent years as will be discussed in the next section. 

A concurrently operated vertical flow scrubber removes the restriction of power input with respect to hydraulic flooding of the packed bed. Therefore, this device can operate at higher gas velocities and greater pressure drops than a countercurrendy operated scrubber. With increased power input possible, concurrendy operated packed scrubbers are much more effective for the removal of particles from 1- to 3-micron equivalent diameter than countercurrent units. Table 5-2 shows typical size ranges for common particulate emissions [3]. Tobacco smoke has been included in this list to provide a comparison of size with commonly encountered particulates. 

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