Laminar flow region

The discharge coefficient for the screen C with aperture D, is given as a function of screen R nolds number Re = D,(V/d) /[L in Fig. 6-16 for plain square-mesfi screens, Ot = 0.14 to 0.79. This cui ve fits most of the data within 20 percent. In the laminar flow region, Re < 20, the discharge coefficient can be computed from... 

David W. Taylor Model Basin, Washington, September 1953 Jackson, loc. cit. Valentin, op. cit.. Chap. 2 Soo, op. cit.. Chap. 3 Calderbank, loc. cit., p. CE220 and Levich, op. cit.. Chap. 8). A comprehensive and apparently accurate predictive method has been publisned [Jami-alahamadi et al., Trans ICE, 72, part A, 119-122 (1994)]. Small bubbles (below 0.2 mm in diameter) are essentially rigid spheres and rise at terminal velocities that place them clearly in the laminar-flow region hence their rising velocity may be calculated from Stokes law. As bubble size increases to about 2 mm, the spherical shape is retained, and the Reynolds number is still sufficiently small (<10) that Stokes law should be nearly obeyed. 

Figure 7-20. A-faotor versus Reynolds number in the laminar flow region 10 < Npg < 2,000. (Source Chen, S. J., Kenics technical data KTEK-2, 1978.)...

NPei and NRtt are based on the equivalent sphere diameters and on the nominal velocities ug and which in turn are based on the holdup of gas and liquid. The Schmidt number is included in the correlation partly because the range of variables covers part of the laminar-flow region (NRei < 1) and the transition region (1 < NRtl < 100) where molecular diffusion may contribute to axial mixing, and partly because the kinematic viscosity (changes of which were found to have no effect on axial mixing) is thereby eliminated from the correlation. 

The power curve for the standard tank configuration is linear in the laminar flow region AB with a slope of —1.0. Thus in this region for ReM< 10, equation 5.20 can be written as... 

For each value of ReM and N in the laminar flow region calculate the apparent viscosity xa from equation 5.14 rewritten in the form... 

In laminar flow region, the breakup time tb and length L corresponding to a small axisymmetric disturbance 8() are expressed as [38]... 

Laminar boundary layer theory assumes that a uniform flow (V = constant) approaches a flat plate. A laminar flow region develops near the plate where the thickness of the laminar boundary layer increases with thickness along the plate, as developed in Example 4.2. If we assign 5 to be the boundary layer thickness, or the distance from the plate where the velocity is equal to 0.99 times the velocity that approached the plate, and 5c to be the concentration boundary layer thickness, then we can see that both 5 and 5c are functions of distance, x, from the leading edge, as shown in Figure 8.11. 

A deeper insight into the physical nature of the process. By presenting experimental data in a dimensionless form, one distinct physical state can be isolated from the other (e.g., turbulent or laminar flow region) and the effect of individual physical variables can be identified. 

Newtonian fluids are those which exhibit a direct proportionality between shear stress and shear rate in the laminar-flow region. This is... 

In general, problems of turbulent flow have been too difficult to study mathematically, at least in their initial stages. If, therefore, one is to develop a relationship between pressure drop and flow rate which is valid for the turbulent- as well as the laminar-flow region, the approach must be empirical—and that of dimensional analysis would seem to be eminently suited to the problem at hand. 

It is thus possible to calculate theoretically the relationship between the friction factor and Reynolds and Hedstrom numbers in the laminar-flow region. Beyond the laminar-flow region the relationships between these three dimensionless groups must be determined experimentally. 

The rigor of this development requires that all fluid-flow data6 follow the conventional / = 16/Nr, relationship in the laminar-flow region. Accordingly, deviations from the theoretical curve in this region are due entirely to errors in measurement or calculation the figures show that these errors were always low except at the lowest Reynolds numbers, where both the flow rates and pressure drops were extremely difficult to determine accurately. 

The end of the stable laminar-flow region usually occurs near... 

As in the case of Newtonian fluids (R7) one would expect that the end of the stable laminar-flow region (generalized = 2100) should not be influenced by the roughness of the pipe. Until experimental data are available, this assumption is recommended although a contrary opinion has been published (W4). 

In the range Re < 20, the proportionality Ne Re is found, thus resulting in the expression NeRe = P/(p,n d ) = const. Density is irrelevant here—we are dealing with the laminar flow region. 

Understandably, the baffles do not influence the power characteristics within the laminar flow region, where viscosity forces prevent rotation of the hquid. However, their influence is extremely strong at Re > 5 X IO". Here, the installation of baffles under otherwise unchanged operating conditions increases the power consumption of the stirrer by a factor of 20 ... 

Clearly, the solution of this equation at forced-convection electrodes will depend on whether the fluid flow is laminar, in the transition regime, or turbulent. Since virtually all kinetic investigations have been performed in the laminar flow region, no further mention will be made of turbulent flow. The reader interested in mass transport under turbulent flow is recommended to consult refs. 14 and 15. 

As might be expected, the dispersion coefficient for flow in a circular pipe is determined mainly by the Reynolds number Re. Figure 2.20 shows the dispersion coefficient plotted in the dimensionless form (Dl/ucI) versus the Reynolds number Re — pud/p(2Ai). In the turbulent region, the dispersion coefficient is affected also by the wall roughness while, in the laminar region, where molecular diffusion plays a part, particularly in the radial direction, the dispersion coefficient is dependent on the Schmidt number Sc(fi/pD), where D is the molecular diffusion coefficient. For the laminar flow region where the Taylor-Aris theory18,9,, 0) (Section 2.3.1) applies ... 

For the laminar flow region of Newtonian fluid, shear stress r is equal to the viscosity /t times the velocity gradient du/dy as... 

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