Particle size dependence reynolds number

While many process fluids are Newtonian, some are non-Newtonian (as seen in Figure 9.7). For such cases, it is not sufQcient to use a single value for viscosity to determine the impeller Reynolds number. Concentrated slurries are typically non-Newtonian and particle-size dependent. They are frequently shear-thinning. Dilatant behavior seldom occurs. If the system is simply shear thinning, it is usually possible to describe its rheological behavior with a simple power-law relationship ... 

The relationship between adsorption capacity and surface area under conditions of optimum pore sizes is concentration dependent. It is very important that any evaluation of adsorption capacity be performed under actual concentration conditions. The dimensions and shape of particles affect both the pressure drop through the adsorbent bed and the rate of diffusion into the particles. Pressure drop is lowest when the adsorbent particles are spherical and uniform in size. External mass transfer increases inversely with d (where, d is particle diameter), and the internal adsorption rate varies inversely with d Pressure drop varies with the Reynolds number, and is roughly proportional to the gas velocity through the bed, and inversely proportional to the particle diameter. Assuming all other parameters being constant, adsorbent beds comprised of small particles tend to provide higher adsorption efficiencies, but at the sacrifice of higher pressure drop. This means that sharper and smaller mass-transfer zones will be achieved. 

The Peclet number Pe = v /Dc, where Dc is the diffusion coefficient of a solute particle in the fluid, measures the ratio of convective transport to diffusive transport. The diffusion time Tp = 2/D< is the time it takes a particle with characteristic length to diffuse a distance comparable to its size. We may then write the Peclet number as Pe = xD/xs, where xv is again the Stokes time. For Pe > 1 the particle will move convectively over distances greater than its size. The Peclet number can also be written Pe = Re(v/Dc), so in MPC simulations the extent to which this number can be tuned depends on the Reynolds number and the ratio of the kinematic viscosity and the particle diffusion coefficient. 

The latter three factors are only relevant for the mass transfer if the Reynolds number (Re = p vr db / q) of the liquid flow around the particle is larger than 1. The size of the gas bubbles depends on liquid properties such as temperature, surface tension and viscosity but also on the dissipated power. If we have to deal with small gas bubbles in a bubble column than we can consider the gas bubbles as rigid. The mass transfer coefficient k q is then given by the equation ... 

The fu st term is a modified Archimedes number, while the second one is the Froude number based on particle size. Alternatively, the first term can be substituted by the Reynolds number. To attain complete similar behavior between a hot bed and a model at ambient conditions, the value of each nondimensional parameter must be the same for the two beds. When all the independent nondimensional parameters are set, the dependent parameters of the bed are fixed. The dependent parameters include the fluid and particle velocities throughout the bed, pressure distribution, voidage distribution of the bed, and the bubble size and distribution (Glicksman, 1984). In the region of low Reynolds number, where viscous forces dominate over inertial forces, the ratio of gas-to-solid density does not need to be matched, except for beds operating near the slugging regime. 

The presence of particles makes the effective conductivity of a gas greater than the molecular conductivity by a factor of 10 or more. The nature of the solid has little effect at Reynolds numbers above 100 or so although the effect is noticeable at the lower values of Re, it has not been completely studied. Besides the Reynolds, Prandtl, and Peclet numbers, the effective diffusivity depends on the molecular conductivity, porosity, particle size, and flow conditions. Plots in terms of Re, Pr, and Pe (without showing actual data points) are made by Beek (1962, Fig. 3), but the simpler plots obtained by a number of investigators in terms of the Reynolds number alone appear on Figure 17.36(a). As Table 17.15 shows, most of the data were obtained with air whose Pr = 0.72 and... 

The liquid-solid mass-transfer coefficient depends mainly on the agitation speed, the particle size, and the physical properties of the system. While ks oc N°-2 this relationship may depend on the particle size (Sano et al., 1974). In a dimensionless form, Sh oc RemSc0 5 however, the value of m changes at some critical Reynolds number when all particles are suspended. The most generalized relationship is given by Eq. (3.34), and its use is recommended. 

In case 3 the relative size of the particles (with respect to the computational cells) is large enough that they contain many hundreds or even thousands of computational cells. It should be noted that the geometry of the particles is not exactly represented by the computational mesh and special, approximate techniques (i.e., body force methods) have to be used to satisfy the appropriate boundary conditions for the continuous phase at the particle surface (see Pan and Banerjee, 1996b). Despite this approximate method, the empirically known dependence of the drag coefficient versus Reynolds number for an isolated sphere could be correctly reproduced using the body force method. Although these computations are at present limited to a relatively low number of particles they clearly have their utility because they can provide detailed information on fluid-particle interaction phenomena (i.e., wake interactions) in turbulent flows. 

They required that the turbulence should be locally isotropic and steady, that the particle Reynolds number should be small, that the particles concentration was small, and that the particle diameter should be much smaller than the length scale of the energy containing eddies for the diffusion controlled range of the model. The model is based on the ability of the particle to respond to the motion of the surrounding fluid. It depends on particle size and density, turbulence structure of the fluid, and transversal particle concentration differences. 

Effect of particle size. The size of the packing influences the Reynolds number and therefore influences the contributions of dispersion and external mass transfer. The contribution from internal diffusion to the total variance is also affected by the particle size. An increase in particle size from 10 pm to 40 pm increases the contribution from external mass transfer more than the other factors investigated here and this influence is dependent upon solute size. Solute displacement is not... 

Table 3-17 gives the Reynolds number, friction factor, and pressure drop of catalyst pellets of 0.25 inch and at different particle length. Table 3-18 shows a typical input data and computer output with PL = 0.25 inch. The simulation exercise gives a pressure drop of 68.603 Ib/in. The results show that the pressure drop in a packed bed depends on size and shape of the particles. 

It is important to remind the reader that U is the velocity of the fluid phase seen by the particle, U - U is the slip velocity, dp is the particle diameter, and Vf is the kinematic viscosity of the fluid phase. Note that Eq. (5.33) depends on the particle velocity U and is valid in the zero-Stokes-number limit where U = U so that particles follow the fluid. The correlation in Eq. (5.31) is valid only for RCp < 1 and Sc > 200. For larger particle Reynolds numbers the following correlations can be used Sh = 2 -i- 0.724Rep Sc, which is valid for 100 < RCp < 2000, and Sh = 2 -i- 0.425RCp Sc, which is valid for 2000 < RCp <10. Among the other correlations available, it is important to cite the one proposed by Ranz Marshall (1952) for macroparticles Sh = 2.0 -i- O.bReJ Sc. These expressions assume that the fluid velocity U is known. For micron-sized (or smaller) particles moving in turbulent fluids for which only the ensemble-mean fluid velocity (Uf) is known, it is instead better to employ the mesoscale model derived by Armenante Kirwan (1989) Sh = 2.0 -i- 0.52(Re ) Sc, where Re = is the modi-... 

Particle diameter, dp, and pressure drop. For a given column performance, the pressure drop is lowest when the adsorbent particles are spherical and of closely uniform size. The external mass-transfer rate increases. inversely as dpM, and the internal rate increases inversely as dfl. The pressure drop variation will depend upon the Reynolds number, but is... 

A nonspherical particle is generally anisotropic with respect to its hydro-dynamic resistance that is, its resistance depends upon its orientation relative to its direction of motion through the fluid. A complete investigation of particle resistance would therefore seem to require experimental data or theoretical analysis for each of the infinitely many relative orientations possible. It turns out, however, at least at small Reynolds numbers, that particle resistance has a tensorial character and, hence, that the resistance of a solid particle of any shape can be represented for all orientations by a few tensors. And the components of these tensors can be determined from either theoretical or experimental knowledge of the resistance of the particle for a finite number of relative orientations. The tensors themselves are intrinsic geometric properties of the particle alone, depending only on its size and shape. These observations and various generalizations thereof furnish most, but not all, of the subject matter of this section. 

At the point of incipient fluidisation, the bed voidage, s f, depends on the shape and size range of the particles, but is approximately equal to 0.4 for isometric particles. The minimmn fluidising velocity V f for a power-law fluid in streamline flow is then obtained by substituting s = Smf in equation (5.67). Although this equation applies oifly at low values of the bed Reynolds numbers (<1), this is not usually a limitation at the high apparent viscosities of most non-Newtonian materials. 

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