Taylor number

Furthermore, in Searle cylinder systems, secondary currents occur above certain Taylor numbers, resulting in greater power input than that resulting with Eq. (12). Comparison with experimentally determined Ne numbers shows that the laminar flow range in internally driven cylindrical stirrers in the range of radius ratios r2/ri= 1.05-2 is only valid for Taylor numbers Takrit= <400-200 [38]. Above Ta >Takritsecondary currents in the... 

The results presented here were found by investigations with a special cyUn-der system [45,48]. This system was constructed for an existing Searle viscosimeter (rotation of inner cylinder), such that the gap widths were large in relation to the reference floe diameter of the floccular system used, so that the formation of the floes and their disintegration in the cylinder system are not impaired. For this system, with r2 = 22 mm, rj = 20.04 mm, and Li = 60 mm (r2/ri > 1.098), the following Newton number relationships were determined from the experimental values collected by Reiter [38] for the Taylor number range of 400 < Ta < 3000 used here ... 

The flow domain of TCP can be described by two dimensionless hydrodynamic parameters, corresponding to the rotational speed of the inner cylinder and the imposed axial flow rate the Taylor number, To, and the axial Reynolds number, Re, respectively ... 

Taylor instability, 77 763 Taylor number, 77 747, 23 190 Taylor System, 27 171 Tazarotene, 25 789 Tazettine, 2 87 TBASE database, 72 467 TBCCO films, 23 872 TBTS, 2 550t TCCA, 73 115... 

When the shear rate reaches a critical value, secondary flows occur. In the concentric cylinder, a stable secondary flow is set up with a rotational axis perpendicular to both the shear gradient direction and the vorticity axis, i.e. a rotation occurs around a streamline. Thus a series of rolling toroidal flow patterns occur in the annulus of the Couette. This of course enhances the energy dissipation and we see an increase in the stress over what we might expect. The critical value of the angular velocity of the moving cylinder, Qc, gives the Taylor number ... 

Smith, G. P., and A. A. Townsend. 1982. Turbulent Couette flow between concentric cylinders at large Taylor numbers. J. Fluid Mechanics 123 187-217. 

Fig. 1.2.17 Relation between Taylor number and particle size of monodispersed silica particles.

Because of the strong effects of plate rotations on the rector performance for both RE and PC electrolyzers, the critical design parameters for these reactors are the Taylor number (a2w/4v)0 5 and the Reynolds number (aVf/v). Here a is the gap width between the plate, w the angular velocity of rotation (in radians per second), v the kinematic viscosity of the fluid, and V the velocity in the feed pipe. Since no asymptotic velocity profile is reached for PC, the length of the cell will be an important design parameter in a pump-cell electrolyzer. Detailed mathematical models for RE and PC electrolyzers are given by Thomas et al. (1988), Jansson (1978), Jansson et al. (1978) and Simek and Rousar (1984). 

According to Taylor (1923), the flow instability is observed when the Taylor number exceeds a critical value where Taylor number is defined by geometrical parameters and the speed of rotation ... 

This can be explained by the fact that the flow in the CCTVFR became closer to plug flow as the Taylor number was dropped closer to. Therefore, the steady-state particle number and the steady-state monomer conversion could be arbitrarily varied by simply varying the rotational speed of the inner cylinder. Moreover, no oscillations were observed, and the rotational speed of the inner cylinder could be kept low, so that the possibility of shear-induced coagulation could be decreased. Therefore, a CCTVFR with these characteristics is considered to be highly suitable as a pre-reactor for a continuous emulsion polymerization process. In the case of the continuous emulsion polymerization of VAc carried out with the same CCTVFR, however, the situation was quite different [365]. Oscillations in monomer conversion were observed, and almost no appreciable increase in steady-state monomer conversion occurred even when the rotational speed of the inner cylinder was decreased to a value close to. Why the kinetic behavior with VAc is so different to that with St cannot be explained at present. 

When the Taylor number exceeds a critical value (Tac), a transition from stable Couette flow to vortical Taylor-Couette flow occurs. The critical Taylor number can be calculated by [24] ... 

Figure 8.11 shows the effect of rotation speed for blbation of baker s yeast suspension and E. coU fermentabon broth [25]. In both cases, the increase in rotation speed in the range of 1000-3000 rpm, which corresponds to a range of Ta from about 2000-6000 for the test device, resulted in a substanbal increase in bux with a transition point observed at a rotation speed of 2000 rpm—this corresponds approximately to a Taylor number of 3500, where turbulent Couette bow can be assumed. 

An empirical correlation which expresses the Sherwood number (Sh) as a function of the Taylor number was proposed by Holeschovsky and Cooney [29] as... 

K. Rotating annulus for reverse osmosis For nonvortical flow IVsfc = 2.15 IV. 4j j° V For vortical flow IVsfc = 1.05 IV. 4j jNi [E,S] Nfa = Taylor number = VididN Vi = inner cylinder radius CO = rotational speed, rad/s d = gap width between cylinders [100]... 

Next we turn to the stability of Couette flow for parallel rotating cylinders. This is an important flow for various applications, and, though it is a shear flow, the stability is dominated by the centrifugal forces that arise because of centripetal acceleration. This problem is also an important contrast with the first two examples because it is a case in which the flow can actually be stabilized by viscous effects. We first consider the classic case of an inviscid fluid, which leads to the well-known criteria of Rayleigh for the stability of an inviscid fluid. We then analyze the role of viscosity for the case of a narrow gap in which analytic results can be obtained. We show that the flow is stabilized by viscous diffusion effects up to a critical value of the Reynolds number for the problem (here known as the Taylor number). 

For Taylor numbers exceeding Tc, the flow develops a secondary flow pattern in which ur and uz are both nonozero. A sketch of the stability criteria given by (3-86) is shown in Fig. 3 8. The reader who is interested in a detailed description of the stability analysis that leads to the criterion (3-86) is encouraged to consult Chap. 12 or one of the standard textbooks on hydrodynamic stability theory (see Chandrashekhar [1992] for a particularly lucid discussion of the instability of Couette flows).12... 

Figure 3-8. Stability diagram for Couette flow with 0 < . 2 / -2 i < 1, plotted as Taylor number T versus ih/ The flow is stable for T < Tc and unstable for T > Tc.

Now, following the assumption of Taylor that the principle of exchange of stabilities is satisfied, so that a = 0 at the neutral stability point, the objective of the stability analysis is to obtain nontrivial solutions of (12-147) and (12-148) with a = 0 for various values of a and then determine the minimum value of T as a function of a. This minimum of T is the critical value of the Taylor number for transition to instability. The corresponding value of a is known as the critical wave number and represents the (dimensionless) wave... 

The critical Taylor number according to this approximate result is the minimum of T over the range of possible values for a, i.e., the value of a, where... 

If we substitute in the definition for the Taylor number for this thin-gap approximation, this can be written in the form... 

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