Reynolds number Subject

Pressure Drop. The prediction of pressure drop in fixed beds of adsorbent particles is important. When the pressure loss is too high, cosdy compression may be increased, adsorbent may be fluidized and subject to attrition, or the excessive force may cmsh the particles. As discussed previously, RPSA rehes on pressure drop for separation. Because of the cychc nature of adsorption processes, pressure drop must be calculated for each of the steps of the cycle. The most commonly used pressure drop equations for fixed beds of adsorbent are those of Ergun (143), Leva (144), and Brownell and co-workers (145). Each of these correlations uses a particle Reynolds number (Re = G///) and friction factor (f) to calculate the pressure drop (AP) per... 

Damkdhler (1936) studied the above subjects with the help of dimensional analysis. He concluded from the differential equations, describing chemical reactions in a flow system, that four dimensionless numbers can be derived as criteria for similarity. These four and the Reynolds number are needed to characterize reacting flow systems. He realized that scale-up on this basis can only be achieved by giving up complete similarity. The recognition that these basic dimensionless numbers have general and wider applicability came only in the 1960s. The Damkdhler numbers will be used for the basis of discussion of the subject presented here as follows ... 

Work done by Wiesner [6] is a much more accurate approach. The subject has also been reported on more recently by Simon and Bulskamper [71. They generally agree with Wiesner that the variance of performance with Reynolds number was more true at low value that at high values. The additional influence above a Reynolds number of 10 is not much. It would appear that if a very close guarantee depended on the Reynolds number to get the compressor within the acceptance range (if the Reynolds number was high to begin with), the vendor would be rather desperate. 

Many materials of practical interest (such as polymer solutions and melts, foodstuffs, and biological fluids) exhibit viscoelastic characteristics they have some ability to store and recover shear energy and therefore show some of the properties of both a solid and a liquid. Thus a solid may be subject to creep and a fluid may exhibit elastic properties. Several phenomena ascribed to fluid elasticity including die swell, rod climbing (Weissenberg effect), the tubeless siphon, bouncing of a sphere, and the development of secondary flow patterns at low Reynolds numbers, have recently been illustrated in an excellent photographic study(18). Two common and easily observable examples of viscoelastic behaviour in a liquid are ... 

The functional dependence of jD on Reynolds number has been the subject of study by many investigators [e.g., Thodos and his co-workers (77, 78), and Wilson and Geankoplis (79)]. A variety of equations have been proposed as convenient representations of the experimental data. Many of these correlations also employ the bed porosity (eB) as an additional correlating parameter. This porosity is the ratio of the void volume between pellets to the total bed volume. 

Resistance functions have been evaluated in numerical compu-tations15831 for low Reynolds number flows past spherical particles, droplets and bubbles in cylindrical tubes. The undisturbed fluid may be at rest or subject to a pressure-driven flow. A spectral boundary element method was employed to calculate the resistance force for torque-free bodies in three cases (a) rigid solids, (b) fluid droplets with viscosity ratio of unity, and (c) bubbles with viscosity ratio of zero. A lubrication theory was developed to predict the limiting resistance of bodies near contact with the cylinder walls. Compact algebraic expressions were derived to accurately represent the numerical data over the entire range of particle positions in a tube for all particle diameters ranging from nearly zero up to almost the tube diameter. The resistance functions formulated are consistent with known analytical results and are presented in a form suitable for further studies of particle migration in cylindrical vessels. 

Hydrodynarhic Resistance of Particles at Small Reynolds Numbers Howard Brenner Author Index—Subject Index... 

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...

Water flows in a 2.5-cm-diameter pipe so that the Reynolds number based on diameter is 1500 (laminar flow is assumed). The average bulk temperature is 35°C. Calculate the maximum water velocity in the tube. (Recall that u, = 0.5wo.) What would the heat-transfer coefficient be for such a system if the tube wall was subjected to a constant heat flux and the velocity and temperature profiles were completely developed Evaluate properties at bulk temperature. 

The choice of equation to use for cross flow over cylinders is subject to some conjecture. Clearly, Eq. (6-17) is easiest to use from a computational standpoint, and Eq. (6-21) is the most comprehensive. The more comprehensive relations are preferable for computer setups because of the wide range of fluids and Reynolds numbers covered. For example, Eq. (6-21) has been successful in correlating data for fluids ranging from air to liquid sodium. Equation (6-17) could not be used for liquid metals. If one were making calculations for air either relation would be satisfactory. 

However, this equation is derived subject to the assumption that the disturbance held is localized and/ or spatially periodic. This assumption removes any contribution coming from the nonlinear convection terms. Lin (1955) and Stuart (1963) point out that various estimates of critical Reynolds number obtained by this approach are erroneously too low due... 

While the general treatment of non-Newtonian liquids is beyond the scope of this review, it may be noted that limited work in this area has given results similar to those described for Newtonian liquids. Plots of power number group against Reynolds number can be used if the Reynolds number is modified to include a parameter characterizing the non-Newtonian behavior. Metzner reviews the subject of non-Newtonian flow in a recent publication (M8). 

During the extrusion of polymers different defects and flow instabilities occur at very low Reynolds numbers. The commonly known ones are sharkskin, melt fracture, slip at the wall and cork flow. These defects are of commercial importance, since they often limit the production rate in polymer processing. Many researchers have been interested in the subject, and thorough reviews on flow stability and melt fracture have been written in the last 30 years [1-4]. More recently, two review papers deahng with viscoelastic fluid mechanics and flow stability, were published by Denn [5] and Larson [6]. However, although much work has been done in the field of extrusion distortions, controversy still exists regarding the site of initiation and physical mechanisms of the instabilities. 

An additional complication that occurs with oscillating flow is the existence of several regimes of laminar and turbulent flow that are functions of frequency as well as Reynolds number, as shown in Figure 3 for the case of smooth circular tubes [2]. These flow regimes are the subject of much research [3]. They are shown as a function of the peak Reynolds number Nr,peak and the ratio of channel radius R to the viscous penetration depth Sv This ratio is sometimes referred to as the dynamic Reynolds number and is similar to the Womersley number Wo = D 28y). In the weakly turbulent regime... 

Basset [10], The formulation of extended force expressions which are valid for larger particle Reynolds numbers to enable a more general description have later been the subject of much experimental and theoretical research. The fundamental modeling principles are outlined in the sequel. 

Taneda [149] studied the flow past a sphere at particle Reynolds numbers between 10 and 10 , and found that the wake is not axis3Tnmetric and that it rotates slowly and randomly about the stream-wise axis. The sphere is thus subject to a random side force. 

To specify the velocity field u, we must solve the Navier-Stokes equations subject to the boundary condition (9-160) at infinity. For present purposes, we follow the example of Section C and assume that the Reynolds number, defined here as Re = a2yp/ii, is very small so that the creeping-flow solution for a sphere in shear flow (obtained in Chap. 8, Section B) can be applied throughout the domain in which 6 differs significantly from unity. Hence, from (8-51) and (8-57), we have... 

The cause of large drag in the case of a body like a circular cylinder is the asymmetry in the velocity and pressure distributions at the cylinder surface that results from separation. All bodies in laminar streaming flow at large Reynolds number are subjected to viscous stresses that boundary-layer analysis shows must be... 

The theory of mixing of a passive scalar concentration field subject to advection and diffusion in a high Reynolds number turbulent flow is based on the works of Obukhov (1949) and Corrsin (1951). Consider a statistically stationary state with a large-scale source of scalar fluctuations in the case when both Pe and Re are large. The... 

A nonspherical particle is generally anisotropic with respect to its hydro-dynamic resistance that is, its resistance depends upon its orientation relative to its direction of motion through the fluid. A complete investigation of particle resistance would therefore seem to require experimental data or theoretical analysis for each of the infinitely many relative orientations possible. It turns out, however, at least at small Reynolds numbers, that particle resistance has a tensorial character and, hence, that the resistance of a solid particle of any shape can be represented for all orientations by a few tensors. And the components of these tensors can be determined from either theoretical or experimental knowledge of the resistance of the particle for a finite number of relative orientations. The tensors themselves are intrinsic geometric properties of the particle alone, depending only on its size and shape. These observations and various generalizations thereof furnish most, but not all, of the subject matter of this section. 

---