Dispersion of a ball settling through a quiescent neutrally buoyant suspension

1998 ◽  
Vol 361 ◽  
pp. 309-331 ◽  
Author(s):  
JAMES R. ABBOTT ◽  
ALAN L. GRAHAM ◽  
LISA A. MONDY ◽  
HOWARD BRENNER
Keyword(s):  
Settling Velocity ◽  
Volume Fraction ◽  
Functional Dependence ◽  
Vertical Component ◽  
Transversely Isotropic ◽  
Suspended Particles ◽  
Three Dimensions ◽  
Ball Diameter ◽  
The Mean ◽  
Neutrally Buoyant

Individual falling balls were allowed to settle through otherwise quiescent well-mixed suspensions of non-colloidal neutrally buoyant spheres dispersed in a Newtonian liquid. Balls were tracked in three dimensions to determine the variances in their positions about a mean uniform vertical settling path. The primary experimental parameters investigated were the size of the falling ball and the volume fraction and size of the suspended particles. Unlike the horizontal variances, the vertical variances were found to be affected by short-time deterministic behaviour relating to the instantaneous local configurational arrangement of the suspended particles. For sufficiently long intervals between successive observations, the trajectories of the balls were observed to disperse about their mean settling paths in a random manner. This points to the existence of a Gaussian hydrodynamic dispersivity that characterizes the linear temporal growth of the variance in the position of a falling ball. The functional dependence of these horizontal and vertical dispersivities upon the parameters investigated was established.The dispersivity dyadic was observed to be transversely isotropic with respect to the direction of gravity, with the vertical component at least 25 times larger than the horizontal component. The vertical dispersivity Dˆv (made dimensionless with the diameter of the suspended spheres and the mean settling velocity) was observed to decrease with increasing falling ball diameter, but to decrease less rapidly with concentration than theoretically predicted for very dilute suspensions; moreover, for falling balls equal in size to the suspended spheres, Dˆv increased linearly with increasing volume fraction ϕ of suspended solids.In addition to the above experiments performed on suspensions of spheres, previously published settling-velocity data on the fall of balls through neutrally buoyant suspensions of rods possessing an aspect ratio of 20 were re-analysed, and vertical dispersivities calculated therefrom. (These data, taken by several of the present investigators in conjunction with other researchers, had only been grossly analysed in prior publications to extract the mean settling velocity of the ball, no attempt having been made at the time to extract dispersivity data too.) The resulting vertical dispersivities, when rendered dimensionless with the rod length and mean settling velocity, showed no statistically significant dependence upon the falling-ball diameter; moreover, all other things being equal, these dispersivities were observed to increase with increasing rod concentration.

Random sequential packing simulations in three dimensions for aligned cubes

1989 ◽  
Vol 26 (3) ◽  
pp. 664-670 ◽  
Author(s):  
Douglas W. Cooper
Keyword(s):  
Volume Fraction ◽  
Three Dimensional ◽  
Random Packing ◽  
Limit Problem ◽  
Three Dimensions ◽  
Standard Errors ◽  
Cube Root ◽  
Random Errors ◽  
Infinite Region ◽  
The Mean

This particular three-dimensional random packing limit problem is to determine the mean fraction of a cubic space that would be occupied by aligned, fixed, equalsize cubes, placed at random locations sequentially until no more can be added. No analytical solution has yet been found for this problem. Simulation results for a finite region and finite number of attempts were extrapolated to an infinite number of attempts (N →∞) in an infinite region by multiple linear regression, using volume fraction occupied (F) as a linear combination of the ratio of the length of the small cube sides (S) to the length of the cubic region side (L) and the cube root of the ratio of the region volume to the total volume of cubes tried, (L3/NS3)⅓. These results for random packing in a volume with penetrable walls can be adjusted with a multiplicative correction factor to give the results for impenetrable walls. A total of N = 107 attempts at placement were made for L/S = 20/1 and N = 14 × 106 attempts were made for L/S = 10/1. The results for volume fraction packed are correlated by F = 0.430(±0.008) + 0.966(±0.072)(S/L) – 0.236(±0.029)(L3/NS)⅓. The numbers in parentheses are twice the standard errors of estimate of the coefficients, indicating the 95% confidence intervals due to random errors. This value for the packing density limit, 0.430 ± 0.008, is slightly larger than that given by a conjecture by Palásti [10], 0.4178. Our value is consistent with that obtained by rather different simulation methods by Jodrey and Tory [8], 0.4227 ± 0.0006, and by Blaisdell and Solomon [2], 0.4262.

scholarly journals Turbulent channel flow of dense suspensions of neutrally buoyant spheres

2015 ◽  
Vol 764 ◽  
pp. 463-487 ◽  
Author(s):  
Francesco Picano ◽  
Wim-Paul Breugem ◽  
Luca Brandt
Keyword(s):  
Volume Fraction ◽  
Solid Phase ◽  
Reynolds Shear Stress ◽  
Plane Channel ◽  
Laminar Flows ◽  
Momentum Balance ◽  
Mean Velocity ◽  
Volume Fractions ◽  
The Mean ◽  
Neutrally Buoyant

AbstractDense particle suspensions are widely encountered in many applications and in environmental flows. While many previous studies investigate their rheological properties in laminar flows, little is known on the behaviour of these suspensions in the turbulent/inertial regime. The present study aims to fill this gap by investigating the turbulent flow of a Newtonian fluid laden with solid neutrally-buoyant spheres at relatively high volume fractions in a plane channel. Direct numerical simulation (DNS) are performed in the range of volume fractions ${\it\Phi}=0{-}0.2$ with an immersed boundary method (IBM) used to account for the dispersed phase. The results show that the mean velocity profiles are significantly altered by the presence of a solid phase with a decrease of the von Kármán constant in the log-law. The overall drag is found to increase with the volume fraction, more than one would expect if just considering the increase of the system viscosity due to the presence of the particles. At the highest volume fraction investigated here, ${\it\Phi}=0.2$, the velocity fluctuation intensities and the Reynolds shear stress are found to decrease. The analysis of the mean momentum balance shows that the particle-induced stresses govern the dynamics at high ${\it\Phi}$ and are the main responsible of the overall drag increase. In the dense limit, we therefore find a decrease of the turbulence activity and a growth of the particle induced stress, where the latter dominates for the Reynolds numbers considered here.

scholarly journals Saltation Layer of Particles in Water Flows Related to Transport Stage

2003 ◽  
Vol 34 (4) ◽  
pp. 343-360 ◽  
Author(s):  
B. S. Mazumder ◽  
D. C. Dalal
Keyword(s):  
Particle Velocity ◽  
Settling Velocity ◽  
Functional Dependence ◽  
Vertical Component ◽  
Proposed Model ◽  
Stress Coefficient ◽  
Physical Assumption ◽  
Drag And Lift ◽  
Particle Velocities ◽  
Horizontal Particle

A theoretical model has been developed to determine the maximum saltation layer thickness of sediment particles in water associated with the migration velocity of particle in the bed layer. This is consistent with Owen's (1964) hypothesis for saltation of uniform grain in air. The equation for mean particle velocity at the bed is derived by balancing the horizontal forces acting on the particle in the bed. The modified expression for mean particle velocity includes the effects of drag and lift coefficients, bed shear stress, coefficient of dynamic friction, settling velocity and pivoting angle. The saltation layer model presented here extends a reasonable physical assumption by converting the average horizontal particle velocity to a vertical component of velocity due to collisions with particles resting on the bed. This explicitly shows a functional dependence of saltation height on mean particle velocity and take-off angle. The proposed model has been tested using available experimental data and the agreement with particle velocities and saltation heights is excellent. An interesting outcome is that a quadratic relationship is suggested between the higher transport stage (upper plane bed) and the take-off angle of particle. This shows that the take-off angle decreases with increase in transport stage.

scholarly journals Suspensions of finite-size neutrally buoyant spheres in turbulent duct flow

2018 ◽  
Vol 851 ◽  
pp. 148-186 ◽  
Author(s):  
Walter Fornari ◽  
Hamid Tabaei Kazerooni ◽  
Jeanette Hussong ◽  
Luca Brandt
Keyword(s):  
Reynolds Number ◽  
Slip Velocity ◽  
Volume Fraction ◽  
Secondary Flows ◽  
Duct Flow ◽  
Turbulent Kinetic Energy Budget ◽  
Square Duct ◽  
The Core ◽  
The Mean ◽  
Neutrally Buoyant

We study the turbulent square duct flow of dense suspensions of neutrally buoyant spherical particles. Direct numerical simulations (DNS) are performed in the range of volume fractions $\unicode[STIX]{x1D719}=0{-}0.2$, using the immersed boundary method (IBM) to account for the dispersed phase. Based on the hydraulic diameter a Reynolds number of 5600 is considered. We observe that for $\unicode[STIX]{x1D719}=0.05$ and 0.1, particles preferentially accumulate on the corner bisectors, close to the corners, as also observed for laminar square duct flows of the same duct-to-particle size ratio. At the highest volume fraction, particles preferentially accumulate in the core region. For plane channel flows, in the absence of lateral confinement, particles are found instead to be uniformly distributed across the channel. The intensity of the cross-stream secondary flows increases (with respect to the unladen case) with the volume fraction up to $\unicode[STIX]{x1D719}=0.1$, as a consequence of the high concentration of particles along the corner bisector. For $\unicode[STIX]{x1D719}=0.2$ the turbulence activity is reduced and the intensity of the secondary flows reduces to below that of the unladen case. The friction Reynolds number increases with $\unicode[STIX]{x1D719}$ in dilute conditions, as observed for channel flows. However, for $\unicode[STIX]{x1D719}=0.2$ the mean friction Reynolds number is similar to that for $\unicode[STIX]{x1D719}=0.1$. By performing the turbulent kinetic energy budget, we see that the turbulence production is enhanced up to $\unicode[STIX]{x1D719}=0.1$, while for $\unicode[STIX]{x1D719}=0.2$ the production decreases below the values for $\unicode[STIX]{x1D719}=0.05$. On the other hand, the dissipation and the transport monotonically increase with $\unicode[STIX]{x1D719}$. The interphase interaction term also contributes positively to the turbulent kinetic energy budget and increases monotonically with $\unicode[STIX]{x1D719}$, in a similar way as the mean transport. Finally, we show that particles move on average faster than the fluid. However, there are regions close to the walls and at the corners where they lag behind it. In particular, for $\unicode[STIX]{x1D719}=0.05,0.1$, the slip velocity distribution at the corner bisectors seems correlated to the locations of maximum concentration: the concentration is higher where the slip velocity vanishes. The wall-normal hydrodynamic and collision forces acting on the particles push them away from the corners. The combination of these forces vanishes around the locations of maximum concentration. The total mean forces are generally low along the corner bisectors and at the core, also explaining the concentration distribution for $\unicode[STIX]{x1D719}=0.2$.

Random sequential packing simulations in three dimensions for aligned cubes

1989 ◽  
Vol 26 (03) ◽  
pp. 664-670 ◽  
Author(s):  
Douglas W. Cooper
Keyword(s):  
Volume Fraction ◽  
Three Dimensional ◽  
Random Packing ◽  
Limit Problem ◽  
Three Dimensions ◽  
Standard Errors ◽  
Cube Root ◽  
Random Errors ◽  
Infinite Region ◽  
The Mean

This particular three-dimensional random packing limit problem is to determine the mean fraction of a cubic space that would be occupied by aligned, fixed, equalsize cubes, placed at random locations sequentially until no more can be added. No analytical solution has yet been found for this problem. Simulation results for a finite region and finite number of attempts were extrapolated to an infinite number of attempts (N →∞) in an infinite region by multiple linear regression, using volume fraction occupied (F) as a linear combination of the ratio of the length of the small cube sides (S) to the length of the cubic region side (L) and the cube root of the ratio of the region volume to the total volume of cubes tried, (L 3/NS 3)⅓. These results for random packing in a volume with penetrable walls can be adjusted with a multiplicative correction factor to give the results for impenetrable walls. A total of N = 107 attempts at placement were made for L/S = 20/1 and N = 14 × 106 attempts were made for L/S = 10/1. The results for volume fraction packed are correlated by F = 0.430(±0.008) + 0.966(±0.072)(S/L) – 0.236(±0.029)(L 3/NS)⅓ . The numbers in parentheses are twice the standard errors of estimate of the coefficients, indicating the 95% confidence intervals due to random errors. This value for the packing density limit, 0.430 ± 0.008, is slightly larger than that given by a conjecture by Palásti [10], 0.4178. Our value is consistent with that obtained by rather different simulation methods by Jodrey and Tory [8], 0.4227 ± 0.0006, and by Blaisdell and Solomon [2], 0.4262.

Study of pumping effect in flow-through electrolysers

1983 ◽  
Vol 48 (8) ◽  
pp. 2232-2248 ◽  
Author(s):  
Ivo Roušar ◽  
Michal Provazník ◽  
Pavel Stuhl
Keyword(s):  
Natural Convection ◽  
Balance Equation ◽  
Volume Fraction ◽  
Industrial Applications ◽  
Pumping Effect ◽  
Flow Through ◽  
The Mean ◽  
The Difference

In electrolysers with recirculation, where a gas is evolved, the pumping of electrolyte from a lower to a higher level can be effected by natural convection due to the difference between the densities of the inlet electrolyte and the gaseous emulsion at the outlet. An accurate balance equation for calculation of the rate of flow of the pumped liquid is derived. An equation for the calculation of the mean volume fraction of bubbles in the space between the electrodes is proposed and verified experimentally on a pilot electrolyser. Two examples of industrial applications are presented.

Pituitary dimensions and volume measurements in pregnancy and post partum

1998 ◽  
Vol 39 (1) ◽  
pp. 64-69 ◽  
Author(s):  
H. Dinç ◽  
F. Esen ◽  
A. Demirci ◽  
A. Sari ◽  
H. Resit Gümele
Keyword(s):  
Pituitary Gland ◽  
Matched Control ◽  
Post Partum ◽  
Three Dimensions ◽  
Maximum Height ◽  
Third Trimester ◽  
Normal Pituitary Gland ◽  
Volume Measurements ◽  
Length Width ◽  
The Mean

Purpose: Our purpose was to clarify and further characterize the changes in height, length, width, volume, and shape in the normal pituitary gland and in width in the infundibulum during pregnancy and the first 6 months post partum. Material and Methods: Cranial MR imaging was performed in 78 women who were pregnant in the second or third trimester or who were post partum, and in 18 age-matched control subjects who were not pregnant. Volume measurements were performed in 2 ways; volume 1=1/2xheightxlengthxwidth; and volume 2=area (measured by trackball)xslice thickness Results: Gland volume, height, width, length, and convexity, and infundibular width increased during pregnancy. the highest values were seen during the 3 days immediately post partum. When compared with volunteers, volumes 1 and 2 showed the largest increase (120%) among the parameters. Gland height showed the best correlation (r=0.94, p>0.00001) with gestational age. the mean height of the gland was 8.76 mm in the third trimester. None of the pregnant women had a gland height of above 10 mm during pregnancy. Only 2 subjects had gland heights of 10.04 and 10.2 mm during the 0–3 days post partum. After this first post-partum period of 3 days, the gland size, shape, and volume and the infundibular width returned to normal within 6 months Conclusion: the pituitary gland enlarges in three dimensions throughout pregnancy. During pregnancy, the volume of the gland shows the highest percentage of increase compared to its length, height, and width. the maximum height of the gland does not exceed 10 mm during pregnancy but it may exceed 10 mm during the 3 days immediately post partum.

Insight into the Formation of Ultrafine Nanostructures in Bulk Amorphous Zr54.5 Ti7.5Al10Cu20Ni8

2001 ◽  
Vol 703 ◽  
Author(s):  
André Heinemann ◽  
Helmut Hermann ◽  
Albrecht Wiedenmann ◽  
Norbert Mattern ◽  
Uta Kühn ◽  
...  
Keyword(s):  
Electron Microscopy ◽  
Volume Fraction ◽  
High Volume ◽  
High Resolution Electron Microscopy ◽  
Scattering Data ◽  
Scanning Calorimetry ◽  
Enhanced Diffusion ◽  
Resolution Electron ◽  
The Mean ◽  
Interparticle Interference

ABSTRACTBulk amorphous Zr54.5 Ti7.5Al10Cu20Ni8 is investigated by means of smal-angle neutron scattering (SANS), differential-scanning calorimetry (DSC), high-resolution electron microscopy (HREM) and other methods. The formation of ultrafine nanostructures in the glassy phase is observed and explained by a new model. Structura fluctuations of randomly distributed partialy ordered domains grow during annealing just below the glass transition temperature by local re-ordering. During anneaing the DSC gives evidence for a increasing volume fraction of the localy ordered domains. At high volume fractions of impinging domains a percolation threshold on the interconnected domain boundaries occurs and enhanced diffusion becomes possible. At that stage SANS measurements lead to satistically significant scattering data. The SANS signals are anayzed in terms of a model taking into account spherica particles surrounded by diffusion zones and interparticle interference effects. The mean radius of the nanocrystaline particles is determined to 1 nm and the mean thickness of the depletion zone is 2 nm. The upper limit for the volume fraction after annealing at 653 K for 4hours is about 20 %. Electron microscopy confirms the size and shows that the particle are crystaline.

Suspensions with a tunable effective viscosity: a numerical study

2012 ◽  
Vol 693 ◽  
pp. 345-366 ◽  
Author(s):  
L. Jibuti ◽  
S. Rafaï ◽  
P. Peyla
Keyword(s):  
Effective Viscosity ◽  
Volume Fraction ◽  
Numerical Study ◽  
Spherical Particles ◽  
Dimensionless Number ◽  
Particle Rotation ◽  
Concentrated Suspensions ◽  
Sheared Suspensions ◽  
Neutrally Buoyant ◽  
Universal Behaviour

AbstractIn this paper, we conduct a numerical investigation of sheared suspensions of non-colloidal spherical particles on which a torque is applied. Particles are mono-dispersed and neutrally buoyant. Since the torque modifies particle rotation, we show that it can indeed strongly change the effective viscosity of semi-dilute or even more concentrated suspensions. We perform our calculations up to a volume fraction of 28 %. And we compare our results to data obtained at 40 % by Yeo and Maxey (Phys. Rev. E, vol. 81, 2010, p. 62501) with a totally different numerical method. Depending on the torque orientation, one can increase (decrease) the rotation of the particles. This results in a strong enhancement (reduction) of the effective shear viscosity of the suspension. We construct a dimensionless number $\Theta $ which represents the average relative angular velocity of the particles divided by the vorticity of the fluid generated by the shear flow. We show that the contribution of the particles to the effective viscosity can be suppressed for a given and unique value of $\Theta $ independently of the volume fraction. In addition, we obtain a universal behaviour (i.e. independent of the volume fraction) when we plot the relative effective viscosity divided by the relative effective viscosity without torque as a function of $\Theta $. Finally, we show that a modified Faxén law can be equivalently established for large concentrations.

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