Reynolds number Stokes law

The drag coefficient for rigid spherical particles is a function of particle Reynolds number, Re = d pii/ where [L = fluid viscosity, as shown in Fig. 6-57. At low Reynolds number, Stokes Law gives 24... 

Fig. 7.1 gives a size spectrum of water-borne particles. Particles with diameters less than 10 pm have been called colloids. In soils, the clay-sized and fine silt-sized particles are classified as colloids. Colloids do not dissolve, but instead remain as a solid phase in suspension. Colloids usually remain suspended because their gravitational settling is less than 10 2 cm s 1. Under simplifying conditions (spherical particles, low Reynolds numbers), Stokes law gives for the settling velocity, vs... 

The low Reynolds number, Stokes-type drag law has been written here in terms of the mean translational friction coefficient f since it is assumed that, because of the Brownian motion, all orientations are equally probable. We recall that / is the mean mobility v. The use of a steady drag law for a particle that is changing its velocity rapidly can be shown to be justified on the basis that the velocities are of such small magnitude that the fluid acceleration can be neglected. 

Forveiy thin hquids, Eqs. (14-206) and (14-207) are expected to be vahd up to a gas-flow Reynolds number of 200 (Valentin, op. cit., p. 8). For liquid viscosities up to 100 cP, Datta, Napier, and Newitt [Trans. In.st. Chem. Eng., 28, 14 (1950)] and Siems and Kauffman [Chem. Eng. Sci, 5, 127 (1956)] have shown that liquid viscosity has veiy little effec t on the bubble volume, but Davidson and Schuler [Trans. Instn. Chem. Eng., 38, 144 (I960)] and Krishnamiirthi et al. [Ind. Eng. Chem. Fundam., 7, 549 (1968)] have shown that bubble size increases considerably over that predic ted by Eq. (14-206) for hquid viscosities above 1000 cP. In fac t, Davidson et al. (op. cit.) found that their data agreed veiy well with a theoretical equation obtained by equating the buoyant force to drag based on Stokes law and the velocity of the bubble equator at break-off ... 

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. 

This chapter is organized into two main parts. To give the reader an appreciation of real fluids, and the kinds of behaviors that it is hoped can be captured by CA models, the first part provides a mostly physical discussion of continuum fluid dynamics. The basic equations of fluid dynamics, the so-called Navier-Stokes equations, are derived, the Reynolds Number is defined and the different routes to turbulence are described. Part I also includes an important discussion of the role that conservation laws play in the kinetic theory approach to fluid dynamics, a role that will be exploited by the CA models introduced in Part II. 

If Stokes Law applies for particle Reynolds numbers lip to 0.2, what is the diameter of the largest particle whose behaviour is governed by Stokes Law for this solid and liquid ... 

This expression for the terminal velocity (i.e., the constant velocity that the particle ultimately attains), is called Stokes law. When the Reynolds number is high, say usually greater than of the order of 800, the flow becomes turbulent flow and eddies form. It was Newton... 

However, more general correlations can be found for Kr in Equation 8.3. If in addition to assuming the particles to be rigid spheres, it is also assumed that the flow is in the laminar region, known as the Stoke s Law region, for Reynolds number less than 1 (but can be applied up to a Reynolds number of 2 without much error) ... 

If Stokes law applies when the Reynolds number is less than 0.2, what is the approximate maximum size of particle to which Stokes Law may be applied under these conditions ... 

Equation 3.1, which is known as Stokes law is applicable only at very low values of the particle Reynolds number and deviations become progressively greater as Re increases. Skin friction constitutes two-thirds of the total drag on the particle as given by equation 3.1. Thus, the total force F is made up of two components ... 

Equation 3.43 was obtained for the Stokes law regime. It overestimates the wall effect, however, at higher particle Reynolds number (Re > 0.2). 

This is Stokes law which is valid in the particle Reynolds number range 10 < Re < 0.20 where the Reynolds number is defined by... 

Stokes law is valid for Reynolds numbers below 0.20 which becomes, for equafion 1.53, Ga < 3.6. 

Sutterby (S7) gave a useful tabulation of the viseosity ratio, defined in Eq. (9-8), for relatively low Re and a. These values, intended primarily to correct for departures from Stokes law in falling sphere viscometry, are shown in Fig. 9.6. Reynolds number is defined using the measured Uj and defined in Eq. (9-9). The curve for a = 0 accounts for departures from the creeping flow approximations in an unbounded fluid, and the relative displacement of the other curves indicates the wall effect. 

Viscosity affects the various mechanisms of separation in accordance with the appropriate settling law. Tor instance, viscosity has no effect on terminal velocities in the range where Newton s law applies except as it affects the Reynolds Number which determines which settling law applies. Viscosity does affect the terminal velocity in both the Intermediate law range and Stokes law range as well as help determine the Reynolds Number. As the pressure increases or the temperature decreases the viscosity of the gas increases. Viscosity becomes a large factor in very small particle separation (Intermediate and Stokes law range). 

The vaporization rates and drag coefficients for 2,2,4-trimethylpentane (iso-octane) sprays in turbulent air streams were determined experimentally by Ingebo (40), who reported that the effect of relative velocity on the evaporation rate was represented by the 0.6 power of the Reynolds number and that the drag coefficient varied inversely with the relative velocity of the drops in the spray. By assuming that the evaporation rate was independent of velocity and the drag coefficient for droplets obeyed Stokes s law, the present author derived a mathematical theory for the ballistics of droplets injected into an air stream for which the velocity varied linearly with distance (57) and... 

Above a Reynolds number of the order of magnitude of 1000, bubbles assume a helmet shape, with a flat bottom (Eckenfelder and Barnhart, loc. cit. and Leibson et al., loc. cit.). After bubbles become large enough to depart from Stokes law at their terminal velocity, behavior is generally complicated and erratic, and the reported data scatter considerably. The rise can be slowed, furthermore, by a wall effect if the diameter of the container is not greater than 10 times the diameter of the bubbles, as shown by Uno and Kintner [AlChE 2, 420 (1956) and Collins, J. Fluid Meek, 28(1), 97 (1967)]. Work has... 

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