Models for the Flow Pattern

In the literature there exists a variety of mathematical models intended to predict cyclone velocity distributions. Such distributions are then used in the modeling or prediction of separation performance. For the cyclone separation models, the velocity distribution in  

We start with the radial velocity, which is computed in a very straightforward manner. Near the wall the radial velocity is neglected, and in the surface CS it is assumed uniform, giving  

In reality the radial velocity in CS is not uniform. There occurs a radial, inwardly directed lip flow or hp leakage just below the vortex tube. Thus some portion of the gas tends to short circuit the imaginary cylinder of height Hcs and diameter near the top of the cylinder. Such behavior is one cause of the observed nonideal s-shaped grade-efficiency cmve discussed in Chap. 3. Another cause is mixing brought about by an axial recirculation of the solids. 

The axial velocity is also fairly straight-forward the surface CS is assumed to separate the outer region of downward flow from the inner region of upward flow. The axial velocities in each region is assmned to be uniform over the cross section. This, together with Eq. (4.2.1), allows calculation of the axial velocity from simple geometrical considerations. 

Unlike the radial and axial velocities mentioned above, a variety of approaches have been advanced for computing the cyclone s tangential velocity Vff. We give an account of some of the models for this under separate headings below. 

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The main contribution from the work of Luo [95, 96] was a closure model for binary breakage of fluid particles in fully developed turbulence flows based on isotropic turbulence - and probability theories. The author(s) also claimed that this model contains no adjustable parameters, a better phrase may be no additional adjustable parameters as both the isotropic turbulence - and the probability theories involved contain adjustable parameters and distribution functions. Hagesaether et al [49, 50, 51, 52] continued the population balance model development of Luo within the framework of an idealized plug flow model, whereas Bertola et al [13] combined the extended population balance module with a 2D algebraic slip mixture model for the flow pattern. Bertola et al [13] studied the effect of the bubble size distribution on the flow fields in bubble columns. An extended k-e model was used describing turbulence of the mixture flow. Two sets of simulations were performed, i.e., both with and without the population balance involved. Four different superficial gas velocities, i.e., 2,4,6 and 8 (cm/s) were used, and the superficial liquid velocity was set to 1 (cm/s) in all the cases. The population balance contained six prescribed bubble classes with diameters set to = 0.0038 (m), d = 0.0048 (m), di = 0.0060 (m), di = 0.0076 (m), di = 0.0095 (m) and di = 0.0120 (m). 

S.D. Kolev, E. Pungor, Description of an axially-dispersed plug flow model for the flow pattern in elements of fluid systems, Anal. Chim. Acta 185 (1986) 315. 

Efficiencies can be scaled up from laboratory data taken with an Oldershaw column (a laboratory-scale sieve-tray column) tFair et al.. 1983 Kister. 1990T The overall efficiency measured in the Oldershaw column is often very close to the point efficiency measured in the large commercial column. This is illustrated in Figure 10-15. where the vapor velocity has been normalized with respect to the fraction of flooding IFair et al 19831. The point efficiency can be converted to Murphree and overall efficiencies once a model for the flow pattern on the tray has been adopted (see section 16.6T... 

Models for cyclone pressure drop sometimes spring from models for the flow pattern and are based on an estimation of the actual dissipative losses in the cyclone others are purely empirical. 

This completes our treatment of the models for the flow pattern in the cyclone. We discuss one more model for vg, namely that of Meissner and Loffler (1978), in Appendix 4.B. We now examine some of the most used pressure drop models. 

Numerical solutions of the maximum mixedness and segregated flow equations for the Erlang model have been obtained by Novosad and Thyn (Coll Czech. Chem. Comm., 31,3,710-3,720 [1966]). A few comparisons are made in Fig. 23-14. In some ranges of the parameters n or fte ihe differences in conversion or reaclor sizes for the same conversions are substantial. On the basis of only an RTD for the flow pattern, perhaps only an average of the two calculated extreme performances is justifiable. 

Simulation models are essential tools for reactor design and optimization. A general simulation model consists of a reactor and a reaction model [1]. The reactor model accounts for the reactor type and for the flow pattern in the reactor, while the reaction or kinetic model describes the kinetics of the chemical reactions occurring. 

Barnea, D., 1987, A Unified Model for Predicting Flow Pattern Transitions for the Whole Range of Pipe Inclinations, Int. J. Multiphase Flow 13 1. (3)... 

Fig. 4.26 Models for the SANS patterns for FCC micellar crystals in Fig. 4.27 (McConnell et al. 1995). (a) Faulted FCC crystal (with equal numbers of FCC and HCP sequences) (b) coexistence between (111) planes (20%) and polycrystalline phase (80%) (c) coexistence between (111) planes (25%) and polycrystalline phase (75%) (d), (e), (f) slipping planes hopping from registry sites with increasing shear-induced flow of layers of micelles. The wavevector range is the same as for Fig. 4.27.

The above formulas are provided as theoretical guidance for the use of the dispersion model. For evaluation of actual coefficients the reader can consult the numerous experimental studies and correlations for tubes, packed and fluidized beds presented by Wen and Fan (58). One should remember that theory only justifies the use of the axial dispersion model at large Peclet nuu ers (Pe >> 1) and at small intensities of dispersion, i.e. D /uL < 0.15. Therefore, attempts in the literature to apply the dispersion model to small deviations from stirred tank behavior, i.e. for large intensities of dispersion, D /uL > 1, such as in describing liquid backmixing in bubble columns, should be considered with caution. Better physical models of the flow patterns are necessary for such situations and the dispersion model should be avoided. 

The basis for the flow pattern is the wave propagation theory as known with light waves or flow waves in physics (Figure 2.23). While in simple models the flow... 

According to the idealized flow patterns (Hlavacek and Kubicek, 1972), the slice of the bed cut in the radial direction can be considered as a fixed-bed with the inlet condition at r = Rx and the outlet condition at r — Rz- Conservation equations written for this slice should be applicable to the whole length of the bed. Therefore, the conservation equations of l s. 9.27 through 9.36 are applicable to radial flow reactors with the convection terms 9c/dz and bT/bz replaced by bc/br and bT/br. This model is then identical to the one-dimensional model with axial dispersion as for the usual fixed-bed reactor. The same boundary conditions as those of Eqs. 9.10 and 9.11 apply, but this time Eq. 9.10 applies at r = Ri and Eq. 9.11 at r = R2. with z replaced by r for the flow pattern (a) in Figure 9.12. Unless the bed depth is quite shallow, the dispersion term can be neglected, resulting in a plug-flow model in the radial direction. 

But, with the above ideas about the nature of cyclone pressiu-e drop and some simplistic considerations, we can see intuitively, as shown below, that cyclone pressure drop, indeed, should decrease with increasing wall fraction in the body (we will also present a macroscopic quasi-theoretical model of cyclone performance with predicts such effects later in this book). Consider two extremes for the flow pattern in the cyclone body ... 

Fig. 15. (a) Values of the flow-pattern efficiency for the two-sheU model, (b) The dependence of the flow-pattern efficiency on the dimensionless... 

Dispersion In tubes, and particiilarly in packed beds, the flow pattern is disturbed by eddies diose effect is taken into account by a dispersion coefficient in Fick s diffusion law. A PFR has a dispersion coefficient of 0 and a CSTR of oo. Some rough correlations of the Peclet number uL/D in terms of Reynolds and Schmidt numbers are Eqs. (23-47) to (23-49). There is also a relation between the Peclet number and the value of n of the RTD equation, Eq. (7-111). The dispersion model is sometimes said to be an adequate representation of a reaclor with a small deviation from phig ffow, without specifying the magnitude ol small. As a point of superiority to the RTD model, the dispersion model does have the empirical correlations that have been cited and can therefore be used for design purposes within the limits of those correlations. 

The objeetive of the following model is to investigate the extent to whieh Computational Fluid Mixing (CFM) models ean be used as a tool in the design of industrial reaetors. The eommereially available program. Fluent , is used to ealeulate the flow pattern and the transport and reaetion of ehemieal speeies in stirred tanks. The blend time predietions are eompared with a literature eonelation for blend time. The produet distribution for a pair of eompeting ehemieal reaetions is eompared with experimental data from the literature. 

The flow pattern is ealeulated from eonservation equations for mass and mometum, in eombination with the Algebraie Stress Model (ASM) for the turbulent Reynolds stresses, using the Fluent V3.03 solver. These equations ean be found in numerous textbooks and will not be reiterated here. Onee the flow pattern is known, the mixing and transport of ehemieal speeies ean be ealeulated from the following model equation ... 

V3.03. The tank diameter was T = 1 m. Furthermore, Z/T = 1, D/T = 0.33, C/T = 0.32, and rpm = 58. The flow pattern in this tank is shown in Figure 10-9. Experimental data were used as impeller boundary eonditions. Figure 10-10 shows the uniformity of the mixture as a funetion of time. The model predietions are eompared with the results of the experimental blend time eorrelation of Fasano and Penny [6]. This graph shows that for uniformity above 90% there is exeellent agreement between the model predietions and the experimental eorrelation. Figure 10-1 la shows the eoneentration field at t = 0 see. Figures 10-1 lb through 10-1 Id show the eoneentration field at t = 0,... 

However, the correlation between and is essentially dependent on the flow pattern, and therefore the correlations, for example Eq. (14.72), are limited to distinctly specified cases. Figure 14.9 illustrates different types of vertical flow, each of which requires its own model for the correlation between and w so-... 

Gal-Or and Resnick (Gl) have developed a simplified theoretical model for the calculation of mass-transfer rates for a sparingly soluble gas in an agtitated gas-liquid contactor. The model is based on the average gas residencetime, and its use requires, among other things, knowledge of bubble diameter. In a related study (G2) a photographic technique for the determination of bubble flow patterns and of the relative velocity between bubbles and liquid is described. 

Steam-liquid flow. Two-phase flow maps and heat transfer prediction methods which exist for vaporization in macro-channels and are inapplicable in micro-channels. Due to the predominance of surface tension over the gravity forces, the orientation of micro-channel has a negligible influence on the flow pattern. The models of convection boiling should correlate the frequencies, length and velocities of the bubbles and the coalescence processes, which control the flow pattern transitions, with the heat flux and the mass flux. The vapor bubble size distribution must be taken into account. 

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