A solid sphere falling through a Bingham plastic moves in a small envelope of fluid with shape that depends on the yield stress. A finite-element/Newton method is presented for solving the free-boundary problem composed of the velocity and pressure fields and the yield surfaces for creeping flow. Besides the outer surface, solid occurs as caps at the front and back of the sphere because of the stagnation points in the flow. The accuracy of solutions is ascertained by mesh refinement and by calculation of the integrals corresponding to the maximum and minimum variational principles for the problem. Large differences from the Newtonian values in the flow pattern around the sphere and in the drag coefficient are predicted, depending on the dimensionless value of the critical yield stress Yg below which the material acts as a solid. The computed flow fields differ appreciably from Stokes’ solution. The sphere will fall only when Yg is below 0.143 For yield stresses near this value, a plastic boundary layer forms next to the sphere. Boundary-layer scalings give the correct forms of the dependence of the drag coefficient and mass-transfer coefficient on yield stress for values near the critical one. The Stokes limit of zero yield stress is singular in the sense that for any small value of Yg there is a region of the flow away from the sphere where the plastic portion of the viscosity is at least as important as the Newtonian part. Calculations For the approach of the flow field to the Stokes result are in good agreement with the scalings derived from the matched asymptotic expansion valid in this limit.

Creeping motion of a sphere through a Bingham plastic

Mass transfer in the continuous phase around axisymmetric bodies of revolution

Empirical correlations for the prediction of mass transfer coefficients for single drops are presented. Published experimental results for both circulating and oscillating drops are considered. Correlation for the individual continuous-phase mass transfer coefficient, which is based on data from 596 measurements taken from 10 different groups of investigators, reproduces the data with an average absolute error of 14.1%. This is then used to determine a correlating equation for the individual dispersed-phase mass transfer coefficient on the basis of data for overall dispersed-phase mass transfer coefficient taken from 21 sources. The average absolute value of the relative error in the predicted values of overall dispersed-phase mass transfer coefficient from the experimental points by using the correlations for individual mass transfer coefficients is 24.5%. It is further shown that by allowing for the effects of power input per unit mass and dispersed-phase hold-up, the correlations for single drops can be extended to extraction columns. The correction factors required for this purpose have been obtained by using simulated values of overall mass transfer coefficients for pulsed perforated-plate, Karr reciprocating-plate, Kuhni, and rotating disc columns.

Correlations for Prediction of Mass Transfer Coefficients in Single Drop Systems and Liquid–Liquid Extraction Columns

Heat transfer, a review of current literature

Approximate velocity profiles and terminal velocities have been predicted for fluid spheres moving in viscous media under the influence of gravity. The equations of motion were satisfied approximately using a numerical method after Kawaguti(1).



Predicted terminal velocities are compared with literature data and with data measured by one of the authors(2). The agreement is reasonable for Reynolds numbers up to about 80.



The effect of the asymmetry of the exterior flow upon the symmetry of the interior flow is discussed.



Les auteurs predisent les profils de vitesse approximatifs et les vitesses terminales pour des spheres fluides qui se deplacent dans un milieu visqueux, sous l'influence de la gravite. On satisfait de facon approximative aux equations du mouvement en utilisant une methode numerique proposee par Kawaguti.1



On compare les vitesses terminales predites avec les donnees de la litterature et avec les valeurs mesurees par un des auteurs.2 L'accord est satisfaisant pour des nombres de Reynold allant jusqu'a 80.



On discute l'effet de l'asymetrie de l'ecoulement exterieur sur la symetrie de l'ecoulement interieur.

Viscous flow around fluid spheres at intermediate reynolds numbers (II)

Approximate local and overall Sherwood numbers have been predicted theoretically for forced-convection transfer around solid and fluid spheres for Reynolds numbers up to 100. The diffusion equation was solved analytically for the case of a thin concentration boundary layer (NPo » 1).



The predicted overall Sherwood numbers are in fair agreement with published semi-empirical correlations.



On predit theoriquement les valeurs approximatives, locales et globales, des nombres de Sherwood pour le transfert par convection forcee autour de spheres solides et fluides, a des nombres de Reynold allant jusqu'a 100. Les auteurs resolvent analytiquement l'equation de diffusion dans le cas d'une couche limite mince (NPo » 1).



Les valeurs globales predites pour les nombres de Sherwood se comparent assez bien a celles que donnent les expressions semi-empiriques deja publiees.

M. H. I. Baird

Papers

Forced convection transfer around spheres at intermediate reynolds numbers