Reynolds number disperse phase

For hquid systems v is approximately independent of velocity, so that a plot of JT versus v provides a convenient method of determining both the axial dispersion and mass transfer resistance. For vapor-phase systems at low Reynolds numbers is approximately constant since dispersion is determined mainly by molecular diffusion. It is therefore more convenient to plot H./v versus 1/, which yields as the slope and the mass transfer resistance as the intercept. Examples of such plots are shown in Figure 16. 

The terms Jga and Jsa are the diffusive fluxes of species a in the gas and solid phases, respectively. Note that in addition to molecular-scale diffusion, these terms include dispersion due to particle-scale turbulence. The latter is usually modeled by introducing a gradient-diffusion model with an effective diffusivity along the lines of Eqs. (149) and (151). Thus, for large particle Reynolds numbers the molecular-scale contribution will be negligible. The term Ma is the... 

The basic assumptions implied in the homogeneous model, which is most frequently applied to single-component two-phase flow at high velocities (with annular and mist flow-patterns) are that (a) the velocities of the two phases are equal (b) if vaporization or condensation occurs, physical equilibrium is approached at all points and (c) a single-phase friction factor can be applied to the mixture if the Reynolds number is properly defined. The first assumption is true only if the bulk of the liquid is present as a dispersed spray. The second assumption (which is also implied in the Lockhart-Martinelli and Chenoweth-Martin models) seems to be reasonably justified from the very limited evidence available. 

Reynolds number corresponding to minimum Co, Table 10.1 Reynolds number based on dispersed phase properties, = Cd/Vp or Ud /Vp Reynolds number corresponding to onset of separation Reynolds number at terminal velocity = Ujd/v... 

The correlation was developed for five 50-mm SMV elements (dH = 8 mm) and covered Reynolds numbers (ReH) in the range 200-20,000 and dispersed-phase volume fraction up to 0.25. 

In order to achieve simultaneous suspension of solid particles and dispersion of gas, it is necessary to define the state when the gas phase is well dispersed. Nienow (1975) defined this to be coincident with the minimum in Power number, Ne, against the aeration number, 1VA, relationship (see Fig. 12 [Sicardi et al., 1981]). While Chapman et al. (1981) accept this definition, their study also showed that there is some critical particle density (relative to the liquid density) above which particle suspension governs the power necessary to achieve a well-mixed system and below which gas dispersion governs the power requirements. Thus, aeration at the critical stirrer speed for complete suspension of solid particles in nonaerated systems causes partial sedimentation of relatively heavy particles and aids suspension of relatively light particles. Furthermore, there may be a similar (but weaker) effect with particle size. Wiedmann et al. (1980), on the other hand, define the complete state of suspension to be the one where the maximum in the Ne-Ren diagram occurs for a constant gas Reynolds number. 

When two liquids are immiscible, the design parameters include droplet size distribution of the disperse phase, coalescence rate, power consumption for complete dispersion, and the mass-transfer coefficient at the liquid-liquid interface. The Sauter mean diameter, dsy, of the dispersed phase depends on the Reynolds, Froudes and Weber numbers, the ratios of density and viscosity of the dispersed and continuous phases, and the volume fraction of the dispersed phase. The most important parameters are the Weber number and the volume fraction of the dispersed phase. Specifically, dsy oc We 06(l + hip ), where b is a constant that depends on the stirrer and vessel geometry and the physical properties of the system. Both dsy and the interfacial area aL remain unaltered, if the same power per unit volume (P/V) is used in the scale-up. 

Stiegel and Shah34 measured the liquid-phase axial dispersion coefficient in a packed rectangular column. Some details of system conditions used in this study have been described earlier, in Sec. 7-3. The axial dispersion coefficient and the liquid-phase Peclet number were correlated to the gas and liquid Reynolds numbers by the expressions... 

Woodburn55 obtained gas-phase axial dispersion data at very high irrigation rales. He found that, over the range 15 < ReG < 500 and 126 < ReL < 1,321 (Reynolds numbers are defined in Sec. 8-3), the gas-phase Peclet number increased with the superficial gas rate. The data indicated that the gas-phase axial dispersion coefficient ZG was proportional to the gas pore velocity i.e., EZG cc C/G, where n > 1 for loading conditions and 0 < n 1 for subloading conditions. The data in the ranges 600 < ReG < 2,200 and 0 = ReL < 375 were well correlated by a correlation of Dunn ct al.,16 namely,... 

In all the studies described above, only the axial dispersion was considered. Anderson et al.1 measured the radial dispersion for the dispersed water phase in an air-water system. The measurements were carried out in a 30.48-cm-diameter Lucite tube packed with 91.44 cm of 1.27-cm Raschig rings. A continuous source of tracer was used. The radial Peclet number decreased with the increase in both the gas and liquid Reynolds number. Some typical results are shown in Fig. 8-5. The measurements were carried out up to the flooding point. The entire results were correlated graphically, as shown in Fig. 8-6. In Figs. 8-5 and 8-6 the Peclet number was based on the fluid velocity through the column, the nominal packing... 

In a horizontal decanter, dispersed phase drops are being carried along the decanter by the flow of the continuous phase. If the velocity of the two separated layers is more than a few centimeters per second, the shape of the dispersion zone will be distorted by drag, and there will be entraiiunent of drops [21], Therefore, the Reynolds number for both phases must be limited. The effect of Reynolds number on liquid-liquid separation is shown in Table 6.14. This hmitation on the Reynolds number will also be used for the dispersed phase to determine the decanter diameter. The minimum diameter is 10.0 cm (0.328 ft) because of wall effects [19]. 

Parametric sensitivity of the criteria has been presented in the form of stability maps. The particle phase dispersion coefficient has been shown to be the most important parameter governing the stability. It has also been shown that the stability maps can be conveniently and advantageously drawn in terms of particle Reynolds number. 

Agitated dispersions at low impeller speeds or high continuous phase viscosities are in a state of laminar or transition flow. At low impeller Reynolds number, (NRe)T < 15, the flow is laminar around the impeller... 

The reader may be surprised not to And a Reynolds number defined speciflcally for the disperse phase. This is because the disperse-phase viscosity is well defined only for Knp 1 (i.e. the collision-dominated or hydrodynamic regime). In this limit, Vp oc oc Knp/Map so that the disperse-phase Reynolds number would be proportional to Map/Krip when Map < 1. However, in many gas-particle flows the disperse-phase Knudsen number will not be small, even for ap 0.1, because the granular temperature (and hence the collision frequency) will be strongly reduced by drag and inelastic collisions. In comparison, molecular gases at standard temperature and pressure have KUp 1 even though the volume fraction occupied by the molecules is on the order of 0.001. This fact can be... 

Flow separation in the case of a drop is delayed compared with the case of a solid particle, and the vorticity region (wake) is considerably narrower. While in the case of a solid sphere, the flow separates and the rear wake occurs at Re 10 (the number Re is determined by the sphere radius), in the case of a drop there may be no separation until Re = 50. For 1 < Re < 50, numerical methods are widely used. The results of numerical calculations are discussed in [94], For these Reynolds numbers, the internal circulation is more intensive than is predicted by the Hadamard-Rybczynski solution. The velocity at the drop boundary increases rapidly with the Reynolds number even for highly viscous drops, In the limit case of small viscosity of the disperse phase, /3 —> 0 (this corresponds to the case of a gas bubble), one can use the approximation of ideal fluid for the outer flow at Re > 1. 

Let us consider a transient solute concentration field in a liquid outside and inside a spherical drop of radius a moving at a constant velocity U in an infinite fluid medium. We assume that the fluid velocity fields for the continuous and disperse phases are determined by the Hadamard-Rybczynski solution [177, 420], obtained for low Reynolds numbers. The concentration far from the drop is maintained constant and equal to C,. At the initial time f = 0, the concentration outside the drop is everywhere uniform and is equal to C inside the drop, it is also uniform, but is equal to Co-... 

However, the effect of a small perturbation in action-action-angle type flows is quite different. The two-parameter family of invariant cycles coalesce into invariant tori that are connected by resonant sheets defined by the u(h,l2) = 0 condition. The consequence of this is that contrary to action-angle-angle flows in this case a trajectory can cover the whole phase space and no transport barriers exist. Thus, in this type of flows global uniform mixing can be achieved for arbitrarily small perturbations. This type of resonance induced dispersion has been demonstrated numerically in a low-Reynolds number Couette flow between two rotating spheres by Cartwright et al. 

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