Reynolds number lower critical

The Reynolds number, which is directly proportional to the air velocity and the size of the obstacle, is a critical quantity. According to photographs presented elsewhere, a regular Karman vortex street in the wake ot a cylinder is observed only in the range of Reynolds numbers from about 60 to 5000. At lower Reynolds numbers, the wake is laminar, and at higher Reynolds numbers, there is a complete turbulent mixing. 

The difference between the calculated and experimentally determined critical Reynolds number can be explained from the reactor configuration, which consisted of four coils connected by straight tubing section. The straight sections would lower the rela-... 

This observation is supported by experiments carried out with a phosphate buffer and a fluorescein solution for visualization of the mixing process. However, in the experiments there are indications that the critical Reynolds number where the mass transfer enhancement sets in is lower (-7) than predicted by the simulations, a fact which is not well understood. 

In Figure 50, the lower curve for E=3.9 v/cm shows a transition in slope. The flux decreases with decreasing Reynolds numbers until a point is reached where the convective transport of particles toward the membrane is just equal to the electrophoretic migration away from the membrane-i.e. the voltage is now the critical voltage. Further decreases in the Reynolds number will not decrease the flux as there is now no concentration polarization. 

Figure 5.18 shows the only reliable Nui c data available near the critical Reynolds number (XI). Since the data were taken with a side support, there is some effect on the separation and transition angles. Thus the values of Nuj are probably subject to error (R2, R3) although the trend with Re should be correct. At Re = 0.87 x 10 the Shi variation is similar to that shown at lower Re in Fig. 5.17. At Re = 1.76 x 10 the critical transition has already occurred, with the separation bubble accounting for the minimum in Nuj c at 0 — 110°. The maximum in Nuj at 0 = 125° reflects the increased transfer rate in the attached turbulent boundary layer. The local minimum at 0 = 160° is due to final separation. These angles do not agree exactly with those in Fig. 5.11 because of the crossflow support and the fact that angular diffusion shifts the...

An increase in Be indicates a competition between the irreversibilities caused by heat transfer and friction. At high Reynolds numbers, the distribution of Be is relatively more uniform than at lower Re. For a circular Couette device, the Reynolds number (Re = wr2lv) at the transition from laminar to turbulent flow is strongly dependent on the ratio of the gap to the radius of the outer cylinder, 1 — n. The critical Re reaches a value 50,000 at 1 n 0.05. We may control the distribution of the irreversibility by manipulating various operational conditions such as the gap of the Couette device, the Brinkman number, and the boundary conditions. 

Tollmien (1931) calculated a critical Reynolds number (the lowest Re3molds number at which the flow first becomes unstable), details of which can be found in Schlichting (1979). The value obtained for the critical Re5molds number by Tollmien was Rex)crit = = 60,000 - a value much lower... 

In Fig. 2.8, the neutral curves for the above four velocity profiles of Fig. 2.7 are compared. The deciding trend is that as H increases, Rccr decreases. And for flows with > 0, the critical Reynolds numbers are significantly lower. 

Fig. 11. Effect of density difference at various liquid viscosities on particle Reynolds number evaluation at lower critical particle diameter, (a) Solid-liquid fluidized beds [a = 3.0, Cv = f(s), pi = 1000 kg/m ]. (b) Gas-solid fluidized beds [a = 3.0, Cy = /(e), po = 1 kg/m ]. (c) Unified stability map of particle Reynolds number vs density difference for different values of transition hold-up solid-liquid fluidized beds [a = 3.0, Cy = f(s), p-l = 1 mPas, pi = 1000 kg/m ].

One should be careful in applying the foregoing considerations to electrodes of widely different dimensions. For example, if one were to employ the RDE in an industrial cell using, say, an electrode area of 1000 cm, the critical Reynolds number would be reached at the rim of the electrode at a rotation rate of only 30 rpm, and turbulence may have occurred at even lower rotation rates, as pointed out earlier. Such electrodes may still be of practical value in an industrial process, as long as one is aware that mass transport may become turbulent beyond a certain radius and that it will not necessarily be uniform on all parts of the electrode. 

The experimental results for Hm obtained by Reddy et al., showed only limited agreement with values calculated from Eqs. (13) and (85). Although the observed trend of the data with varying particle Rejmolds number, shown in Fig. 29, was correctly predicted, agreement between the calculated and experimental results for Hm was obtained only for those sj tems where the Rejmolds numbers were close to or greater than the critical value of 70. For lower Reynolds numbers, the predicted values were found to be generally too high. 

For aqueous solutions, say, at 25°C the lower limit of the critical Reynolds number will not be reached as long as V/Rt is well below 30. Thus for all practical purposes this complication will not arise in routine viscosity measurements. On the other hand if the non-Newtonian viscosity measurements are extended to a very high range of the shearing stress, it is desirable to check the possibility of this effect. 

If the characteristic linear dimension of the flow field is small enough, then the measured hydrodynamic data differ from those predicted by the Navier-Stokes equations [79]. With respect to the value in macrocharmels, in microchannels (around 50 microns of section) (i) the friction factor is about 20-30% lower, (ii) the critical Reynolds number below which the flow remains laminar is lower (e.g., the change to turbulent flow occurs at lower linear velocities) and (iii) the Nusselt number, for example, heat transfer characteristics, is quite different [80]. The Nusselt number for the microchannel is lower than the conventional value when the flow rate is small. As the flow rate through the microchannel is increased, the Nusselt number significantly increases and exceeds the value for the fully developed flow in the conventional channel. These effects have been investigated extensively in relation to the development of more efficient cooling devices for electronic applications, but have clear implications also for chemical applications. 

Note that in fact the plane Poiseuille flow (1.17) is also an exact solution of the full Navier-Stokes equation. However, it was shown by linear stability analysis that this becomes unstable to small perturbations at a critical Reynolds number of 5772. In fact, the transition to turbulence is observed experimentally at even lower values of Re around 1000. 

With parallel flow along a plate, turbulent flow first appears at a critical Reynolds number between about 10 and 3 x 10. The transition occurs at the lower Reynolds numbers when the plate is rough and the intensity of turbulence in the approaching stream is high and at the higher values when the plate is smooth and the intensity of turbulence in the approaching stream is low. 

Critical Reynolds Number. The Reynolds number, defined as umDhlv, is widely adopted to identify flow status such as laminar, turbulent, and transition flows. A great number of experimental investigations have been performed to ascertain the critical Reynolds number at which laminar flow transits to turbulent flow. It has been found that the transition from laminar flow to fully developed turbulent flow occurs in the range of 2300 Re < 104 for circular ducts [41]. Correspondingly, flow in this region is termed transition flow. More conservatively, the lower end of the critical Reynolds number is set at 2100 in most applications. 

Critical Reynolds Number For concentric annular ducts, the critical Reynolds number at which turbulent flow occurs varies with the radius ratio. Hanks [109] has determined the lower limit of Recrit for concentric annular ducts from a theoretical perspective for the case of a uniform flow at the duct inlet. This is shown in Fig. 5.16. The critical Reynolds number is within 3 percent of the selected measurements for air and water [109]. 

FIGURE 5.16 Lower limits of the critical Reynolds numbers for concentric annular ducts with uniform velocity at the inlet [109]. 

Transition Flow. The lower limit of the critical Reynolds number Recrit for a parallel plate duct is reported to be between 2200 and 3400, depending on the entrance configurations and disturbance sources [143]. The following friction factor formula developed by Hrycak and Andrushkiw [144] is recommended for transition flow in the range of 2200 < Re < 4000 ... 

TABLE 5.37 Lower Limits of the Critical Reynolds Numbers for Smooth Rectangular Ducts... 

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