Wall correction

TABLE 6-9 Wall Correction Factor for Rigid Spheres in Stokes Law Region... 

With cold-wall effect, two cold-wall correction factors, FA and Fc, are used, and the correlation becomes, (Reddy and Fighetti, 1982)... 

Although these wall correction factors appear to be independent of Reynolds number for small (Stokes) and large (> 1000) values of NRe, the value of Kxv is a function of both lVRe and d/D for intermediate Reynolds numbers (Chhabra, 1992). 

The wall effect for particles settling in non-Newtonian fluids appears to be significantly smaller than for Newtonian fluids. For power law fluids, the wall correction factor in creeping flow, as well as for very high Reynolds... 

KWli low Reynolds number wall correction factor, [—]... 

Many of the data on the gross terminal velocity of drops have been taken in vertical cylindrical glass tubes of limited size. To interpret such data in terms of a drop moving in an infinite medium, a wall correction factor is necessary. 

Their solution, involving Gegenbauer functions, resulted in a wall correction factor for rigid spheres which is the same as the full expression corresponding to Eq. (32). 

Using a similar attack for a fully circulating fluid sphere in a stationary field but using only n = 2 due to inconsistency of the equations for higher order functions, their wall correction factor was... 

Fig. 9.2 Wall correction coefficient [C in Eq. (9-10)] for a rigid particle settling eccentrically in a circular cylinder.

Wall Correction Factor K for a Rigid Spherical Particle Moving on the Axis of a Cylindrical Tube in Creeping Flow... 

Fig. 9.3 Wall correction factors K for a rigid sphere on the axis of a cylinder of finite length in creeping flow (S7) (1) L /D = 1 (2) LJD = f (3) LJD = i.

Given the fluid density as a function of pressure, the relative viscosity can be directly determined by the ratio of settling velocities, independently of any determination of the sphere radius or the wall correction. The compressibility of the sphere is negligible over the pressure range used. Furthermore, if the fluid compressibility is unknown, only a relatively small error is made (about 5 per cent for a typical fluid compressibility) in neglecting the density term in eqn (3.9.3). 

To measure the absolute viscosity eqn (3.9.2) must be solved, and so a value of y is needed. The discussion above indicates the conditions necessary to produce a constant wall correction, but there are unknowns, such as the sphere-to-wall separation, which make it impossible to calculate an actual theoretical value with sufficient accuracy. One must measure y empirically by observing the sphere velocity under conditions for which the fluid viscosity is known. For the rolling-ball viscometer, comparison is made with ambient-pressure viscosities whereas high-pressure data taken by the rolling-ball viscometer are used for comparison for the centrifugal-force viscometer. 

Table 3.2. The wall-correction parameter, y, at a given tilt angle, 0, for the rolling-ball method eqn (3.9.2) and /cont for the centrifugal-force method eqn. (3.9.4)...

The 1962 ASME Code Section VIII, Division 1 gives equation 4-4 for thin-walled (f <0.356r) spherical shells, which also applies to hemispherical heads. If the thick-wall correction factor (-0.6p and -0.2p for cylindrical shell and hemispheric head, respectively) is omitted, the thicknesses are 1 2, respectively. 

Fortunately, in most practical investigations, such complete information is unnecessary. Rather, it usually suffices to know only certain components of these dyadics, and then only in limiting cases. If ajl represents a characteristic particle-to-wall dimension ratio, these limiting cases correspond to the extreme cases where ajl is either very small or very near unity. In the former case the method of reflections (cf. H9) provides a useful technique for obtaining the wall correction. In the latter case, corresponding to the situation where the particle is extremely close to the wall, lubrication-theory type approximations (B7, B29, Cll, D7, G5d, H15, K8, M9, MIO, S8) normally suffice to obtain the required correction. Intermediate cases are rarely of interest. 

For example, from the known solution for laminar flow in an elliptical duct (B4, p. 69) of semiaxes b and c one obtains 2a b + c )l ib c as the appropriate wall-correction factor. 

The measured values of polymer flow taken by capillary rheometers are often presented as plots of shear stress versus shear rate at certain temperatures. These values are called apparent shear stress and apparent shear rate at the tube wall. Corrections must be applied to these values in order to obtain true values. The corrected value of shear stress is determined by the Bagley correction [20]... 

On the other hand, through the use of the boundary collocation technique, the diffusiophoretic motion of a colloidal sphere with a thin but polarized diffuse layer in the direction perpendicular to a plane wall was examined. The wall effect on diffusiophoresis was found to be a complicated function of the properties of the particle and ions. The diffusiophoretic motions of a spherical particle with a thin polarized diffuse layer parallel and normal to two plane walls at an arbitrary positiOTi between them and alrnig the axis of a circular cylindrical pore were also investigated by using the boundary collocation method [9]. Numerical results of the wall corrections to Eqs. 5 and 11 for the particle velocity were presented for various values of the relative separation distances and other relevant parameters. 

Currently available limited experimental data do not permit the formulation of generalized predictive expressions even for macroscopic quantities like drag coefficient and wall correction factors. Thus, the empirical correlations presented herein must be regarded as somewhat tentative in their scope, and extrapolations beyond their ranges of applicability must be treated with reserve. Finally, it is hoped that this review will stimulate further interest in this hitherto somewhat neglected field of non-Newtonian fluid/particle mechanics. 

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