High Reynolds Numbers

At high Reynolds numbers the friction factor becomes nearly constant, approaching a value of the order of unity for most packed beds. In terms of S, particle surface area per unit volume of bed,... 

Porous Media Packed beds of granular solids are one type of the general class referred to as porous media, which include geological formations such as petroleum reservoirs and aquifers, manufactured materials such as sintered metals and porous catalysts, burning coal or char particles, and textile fabrics, to name a few. Pressure drop for incompressible flow across a porous medium has the same quahtative behavior as that given by Leva s correlation in the preceding. At low Reynolds numbers, viscous forces dominate and pressure drop is proportional to fluid viscosity and superficial velocity, and at high Reynolds numbers, pressure drop is proportional to fluid density and to the square of superficial velocity. 

A perfect fluid is a nonviscous, noucouducting fluid. An example of this type of fluid would be a fluid that has a very small viscosity and conductivity and is at a high Reynolds number. An ideal gas is one that obeys the equation of state ... 

Curve a is for marine propellers, d-jd = i/i, set a distance C = d or greater from tbe bottom of tbe vessel. The effect of changing d /dj is apparently felt only at very high Reynolds numbers and is not well estab-hsbed. 

Curve g is for disk flat-blade turbines operated in unbaffled vessels filled witb liquid, covered, so tbat no vortex forms. If baffles are present, tbe power characteristics at high Reynolds numbers are essentially tbe same as curve h for baffled open vessels, witb only a slight increase in power. 

Figure 2.2.4 (Berty 1983) shows a tubular reactor that has a thermosiphon temperature control system. The reaction is conducted in the vertical stainless steel tube that can have various diameters, 1/2 in. being the preferred size. If used for fixed bed catalytic studies, it can be charged with a single string of catalytic particles just a bit smaller than the tube, e.g., 5/16 particles in a l/2 O.D. tube. With a smaller catalyst, a tube with an inside diameter of up to three to four particle diameters can be used. With such catalyst charges and a reasonably high Reynolds number— above 500, based on particle diameter—this reactor...

PTC 10 has one correlation that has been found to be incorrect. Equation 5.27-1 will permit a Reynolds number correction for high Reynolds number gas (above 10 ), which is much too optimistic. ISO standards allowed no corrections, which is more nearly coirect. 

Rush ton et al. [10] investigated the effect of varying the tank geometrical ratios and the correlation of the Power number with Reynolds number. At high Reynolds number, it was inferred that. 

The length of the circulation zone (bubble), L, created when the linear jet is supplied at an angle a to the surface was studied experimentally by Bourque and Newman and theoretically by Sawyer.The effect of the angle between the jet axis at the outlet and the surface on the length of the circulation bubble is shown in Fig. 7.32, reproduced from Awbi. The data presented in Fig. 7.32 show that at sufficiently high Reynolds number the length of the circulation zone is independent of the Reynolds number. 

Shekunov, B. Yu., Baldyga, J. and York, P., 2001. Particle formation by mixing with supercritical antisolvent at high Reynolds numbers. Chemical Engineering Science, 56(7), 2421-2433. 

The high Reynolds number represents the turbulent flow regime. 

Chain degradation in turbulent flow has been frequently reported in conjunction with drag reduction and in simple shear flow at high Reynolds numbers [187], Using poly(decyl methacrylate) under conditions of turbulent flow in a capillary tube, Muller and Klein observed that the hydrodynamic volume, [r ] M, is the determining factor for the degradation rate in various solvents and at various polymer concentrations [188], The initial MWD of the polymers used in their experiments are, however, too broad (Mw/Iiln = 5 ) to allow for a precise... 

This relation holds provided that the one-seventh power law may be assumed to apply over the whole of the cross-section of the pipe. This is strictly the case only at high Reynolds numbers when the thickness of the laminar sub-layer is small. By combining equations 3.59 and 3.63, the velocity profile is given by ... 

Thus for turbulent flow at high Reynolds numbers, where the thickness of the laminar sub-layer may be neglected, a 1. 

The relation between the mean velocity and the velocity at the axis is derived using this expression in Chapter 3. There, the mean velocity u is shown to be 0.82 times the velocity us at the axis, although in this calculation the thickness of the laminar sub-layer was neglected and the Prandtl velocity distribution assumed to apply over the whole cross-section. The result therefore is strictly applicable only at very high Reynolds numbers where the thickness of the laminar sub-layer is vety small. At lower Reynolds numbers the mean velocity will be rather less than 0.82 times the velocity at the axis. 

In addition to momentum, both heat and mass can be transferred either by molecular diffusion alone or by molecular diffusion combined with eddy diffusion. Because the effects of eddy diffusion are generally far greater than those of the molecular diffusion, the main resistance to transfer will lie in the regions where only molecular diffusion is occurring. Thus the main resistance to the flow of heat or mass to a surface lies within the laminar sub-layer. It is shown in Chapter 11 that the thickness of the laminar sub-layer is almost inversely proportional to the Reynolds number for fully developed turbulent flow in a pipe. Thus the heat and mass transfer coefficients are much higher at high Reynolds numbers. 

The correlation of Norwood and Metzner shows to be a complex function of the Reynolds number, the Froude number, the ratio of tank-to-impeller diameter, and the ratio of tank diameter to liquid level. However, to a reasonable first approximation for geometrically similar vessels operating at high Reynolds numbers. 

The above analysis is restricted to high Reynolds numbers, although the definition of high is different in a stirred tank than in a circular pipe. The Reynolds number for a conventionally agitated vessel is defined as... 

The experimental results for udp /Dr) in Figure 9.1 show a wide range of values at low Reynolds numbers. The physical properties of the fluid, and specifically its Schmidt number. Sc = /x/(pS) ), are important when the Reynolds number is low. Liquids will lie near the top of the range for u dp /Dr)oo gases near the bottom. At high Reynolds numbers, hydrodynamics dominate, and the fluid properties become unimportant aside from their effect on Reynolds number. This is a fairly general phenomenon and is discussed further... 

The parameter D is known as the axial dispersion coefficient, and the dimensionless number, Pe = uL/D, is the axial Peclet number. It is different than the Peclet number used in Section 9.1. Also, recall that the tube diameter is denoted by df. At high Reynolds numbers, D depends solely on fluctuating velocities in the axial direction. These fluctuating axial velocities cause mixing by a random process that is conceptually similar to molecular diffusion, except that the fluid elements being mixed are much larger than molecules. The same value for D is used for each component in a multicomponent system. 

Because of fhe planar nafure of the cormterflow flame and the relatively high Reynolds number associated with the flow, the flame/flow configuration can be considered to be "aerodynamically clean," where the quasi-one-dimensional and bormdary-layer simplifications can be implemented in either analytical or computational studies. Useful insights into the thermochemical structure... 

Chomiak, ]., Dissipation fluctuations and the structure and propagation of turbulent flames in premixed gases at high Reynolds numbers. Proceedings of the Combustion Institute, 16, 1665-1673,1977. 

M.S. Wu, S. Kwon,J. Driscoll, andG.M. Faeth 1990, Turbulent premixed hydrogen-air flames at high Reynolds numbers. Combust. Set. Technol. 73(l-3) 327-350. 

Clemens, N.T., Paul, P.H., and Mungal, M.G., The structure of OH fields in high Reynolds number turbulent jet diffusion flames. Combust. Sci, Technol., 129,165,1997. 

For effective mixing, high Reynolds numbers are required and the turbulence should be dissipated in the bulk of the flow and as little as possible on the walls. The source term for generating turbulent energy is in the k—s model written as... 

The simulations of fluid flow and heat transfer in such microstructured geometries were carried out with an FVM solver. Air with an inlet temperature of 100 °C was considered as a fluid, and the channel walls were modeled as isothermal with a temperature of 0 °C. The streamline pattern is characterized by recirculation zones which develop behind the fins at comparatively high Reynolds numbers. The results of the heat transfer simulations are summarized in Figure 2.34, which shows the Nusselt number as a fimction of Reynolds number. For... 

Flow of the liquid past the electrode is found in electrochemical cells where a liquid electrolyte is agitated with a stirrer or by pumping. The character of liquid flow near a solid wall depends on the flow velocity v, on the characteristic length L of the solid, and on the kinematic viscosity (which is the ratio of the usual rheological viscosity q and the liquid s density p). A convenient criterion is the dimensionless parameter Re = vLN, called the Reynolds number. The flow is laminar when this number is smaller than some critical value (which is about 10 for rough surfaces and about 10 for smooth surfaces) in this case the liquid moves in the form of layers parallel to the surface. At high Reynolds numbers (high flow velocities) the motion becomes turbulent and eddies develop at random in the flow. We shall only be concerned with laminar flow of the liquid. 

The power number becomes independent on mixing at turbulent conditions, which are achieved at Reynolds numbers greater than 20,000. Typical power numbers at high Reynolds number for some common stirrer types are shown in Table 5.4.22. 

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