Volume-averaged Velocity Research Articles

Modeling of the Dispersed-Phase Size Distribution in Bubble Columns

A population balance model formulation for predicting the size distribution of the dispersed gas phase in bubble column reactors is presented. Extended source term parametrizations for the breakup and coalescence mechanisms are included in a computational fluid dynamics (CFD) model and tested on a fairly simple flow formulation corresponding to locally obtained experimental data. Most CFD codes are based on the assumption of a monodisperse bubble size distribution resulting in an inaccurate prediction of the hold-up. Comparing the results obtained by the breakup model versions against experimental data indicates that the extended model provides an improved description of the bubble size distribution, hold-up, and thus also the volume-averaged gas-phase velocity.

Flow, diffusion, and thermal convection in percolation clusters: NMR experiments and numerical FEM/FVM simulations

Percolation objects were fabricated based on computer-generated, two- or three-dimensional templates. Random-site, semi-continuous swiss cheese, and semi-continuous inverse swiss-cheese percolation models above the percolation threshold were considered. The water-filled pore space was investigated by NMR imaging and, in the presence of a pressure gradient, NMR velocity mapping. The fractal dimension, the correlation length, and the percolation probability were evaluated both from the computer-generated templates and the corresponding NMR spin density maps. Based on velocity maps, the percolation backbones were determined. The fractal dimension of the backbones turned out to be smaller than that of the complete cluster. As a further relation of interest, the volume-averaged velocity was calculated as a function of the probe volume radius. In a certain scaling window, the resulting dependence can be represented by a power law the exponent of which was not yet considered in the theoretical literature. The experimental results favorably compare to computer simulations based on the finite-element method (FEM) or the finite-volume method (FVM). Percolation theory suggests a relationship between the anomalous diffusion exponent and the fractal dimension of the cluster, i.e., between a dynamic and a structural parameter. We examined interdiffusion between two compartments initially filled with H 2O and D 2O, respectively, by proton imaging. The results confirm the theoretical expectation. As a third transport mechanism, thermal convection in percolation clusters of different porosities was studied with the aid of NMR velocity mapping. The velocity distribution is related to the convection roll size distribution. Corresponding histograms consist of a power law part representing localized rolls, and a high-velocity cut-off for cluster-spanning rolls. The maximum velocity as a function of the porosity clearly visualizes the percolation transition.

Flow through percolation clusters: NMR velocity mapping and numerical simulation study.

Three- and (quasi-)two-dimensional percolation objects have been fabricated based on Monte Carlo generated templates. The object size was up to 12 cm (300 lattice sites) in each dimension. Random site, semicontinuous swiss-cheese, and semicontinuous inverse swiss-cheese percolation models above the percolation threshold were considered. The water-filled pore space was investigated by nuclear magnetic resonance (NMR) imaging and, after exerting a pressure gradient, by NMR velocity mapping. The spatial resolutions of the fabrication process and the NMR experiments were 400 microm and better than 300 microm, respectively. The experimental velocity resolution was 60 microm/s. The fractal dimension, the correlation length, and the percolation probability can be evaluated both from the computer generated templates and the corresponding NMR spin density maps. Based on velocity maps, the percolation backbones were determined. The fractal dimension of the backbones turned out to be smaller than that of the complete cluster. As a further relation of interest, the volume-averaged velocity was calculated as a function of the probe volume radius. In a certain scaling window, the resulting dependence can be represented by a power law, the exponent of which was not yet considered in the theoretical literature. The experimental results favorably compare to computer simulations based on the finite-element method (FEM) or the finite-volume method (FVM). This demonstrates that NMR microimaging as well as FEM/FVM simulations reliably reflect transport features in percolation clusters.

Reconstructing cosmic peculiar velocities from the mildly non-linear density field

We present a numerical study of the cosmic density vs. velocity divergence relation (DVDR) in the mildly non-linear regime. We approximate the dark matter as a nonrelativistic pressureless fluid, and solve its equations of motion on a grid fixed in comoving coordinates. Unlike N-body schemes, this method yields directly the volumeaveraged velocity field. The results of our simulations are compared with the predictions of the third-order perturbation theory (3PT) for the DVDR. We investigate both the mean ‘forward’ relation (density in terms of velocity divergence) and the mean ‘inverse’ relation (velocity divergence in terms of density), with emphasis on the latter. On scales larger than about 20 megaparsecs, our code recovers the predictions of 3PT remarkably well, significantly better than recent N-body simulations. On scales of a few megaparsecs, the DVDR predicted by 3PT differs slightly from the simulated one. In particular, approximating the inverse DVDR by a third-order polynomial turns out to be a poor fit. We propose a simple analytical description of the inverse relation, which works well for mildly non-linear scales.

Flow, Diffusion, Dispersion, and Thermal Convection in Percolation Clusters: NMR Experiments and Numerical FEM/FVM Simulations

AbstractBased on computer-generated templates, percolation objects were fabricated. Random-site, semi-continuous swiss cheese, and semi-continuous inverse swiss-cheese percolation models above the percolation threshold were considered. The water-filled pore space was investigated by nuclear magnetic resonance (NMR) imaging and in the presence of a pressure gradient, by NMR velocity mapping. The percolation backbones were determined using velocity maps. The fractal dimension of the backbones turned out to be smaller by about 17 % than that of the complete cluster. As a further relation of interest, the volume-averaged velocity was calculated as a function of the probe volume radius. In a certain scaling window, the resulting dependence can be represented by a power law. The experimental results favorably compare to computer simulations with the finite-element method (FEM) or the finite-volume method (FVM). Thermal convection in percolation clusters of different porosities was studied using the NMR velocity mapping technique. The velocity distribution is related to the convection roll size distribution. The maximum velocity as a function of the porosity clearly visualizes a closed-loop percolation transition if the Rayleigh number conditions are appropriate. Percolation theory suggests a relationship between the anomalous diffusion exponent and the fractal dimension of the cluster,i.e.between a dynamic and a structural parameter. Interdiffusion between two compartments initially filled with H2O and D2O, respectively, was examined by proton imaging. The results confirm the theoretical expectation. Finally, advection driven dispersive transport was investigated in the large Péclet number limit. The superdiffusive transport anomaly was demonstrated and discussed in terms of the non-local advection-diffusion and the fractional diffusion theories.

A computational simulation of an alkaline fuel cell

An advanced one-dimensional, isothermal mathematical model for a single cell of an alkaline fuel cell (AFC) is presented. Advances in an expression for the volume average velocity and in correlations and parameters are achieved. The parameters and operating conditions of the model are based on the Obiter Fuel Cell, which is employed as a power source for NASA space shuttles. A stimulated result is obtained that shows a close agreement with some of the experimental data. Profiles of variables, local overpotential and local current density are also obtained as a function of cell voltage. An investigation of the influence of initial electrolyte concentration shows that the performance of the AFC is maximized at a concentration of 3.5 M. Finally, it is found that increasing the operating pressure steadily enhances cell performance.

Forced Convection in Microstructures for Electronic Equipment Cooling

This paper reports analytical solutions for both velocity and temperature profiles in Microchannel heat sinks by modeling the Microchannel heat sink as a fluid-saturated porous medium. The analytical solutions are obtained based on the modified Darcy model for fluid flow and the two-equation model for heat transfer. To validate the porous medium model and the analytical solutions based on that model, the closed-form solution for the velocity distribution in the fully-developed channel flow and the numerical solutions for the conjugate heat transfer problem, comprising the solid fin and the fluid, are also obtained. The analytical solutions based on the porous medium model are shown to predict the volume-averaged velocity and temperature distributions quite well. Using the analytical solutions, the aspect ratio and the effective thermal conductivity ratio are identified as variables of engineering importance and their effects on fluid flow and heat transfer are studied. As either one of these variables increases, the fluid temperature is shown to approach the solid temperature. Finally, the expression for the total thermal resistance, derived from the analytical solutions and the geometry of the microchannel heat sink for which the thermal resistance of the heat sink is minimal, is obtained.

A Lagrangian Vorticity Method for Two-Phase Particulate Flows with Two-Way Phase Coupling

A Lagrangian vorticity-based method is presented for simulating two-way phase interaction in a two-phase flow with heavy particles. The flow is computed by solving the vorticity transport equation, including the particle-induced vorticity source, and the mass conservation equation for particle concentration on separate sets of fluid control points and particle control points, respectively. The fluid control points are advected with the local fluid velocity, plus a diffusion velocity for viscous problems to account for the spread of the vorticity support via diffusion, while the particle control points are advected by solution of the Lagrangian particle momentum equation. The particle concentration and vorticity transport equations are evaluated using volume-averaged particle velocity and contact force fields, obtained by a weighted average over nearby particle control points. One novel feature of the numerical method is the scheme for calculation of the particle-induced vorticity source using a “moving least-square” differentiation scheme across the two sets of control points. Another feature of the method is its ability to absorb the vorticity generated by particle forces through an adaptive scheme for generation of new fluid control points. Test calculations with a vortex patch filled with particles show that the numerical results compare well with the results obtained both by a traditional finite-difference method and by an asymptotic approximation valid for small Stokes numbers. Other features of the numerical method are demonstrated for calculations involving a particle cloud falling under gravity and a two-phase mixing layer flow.

No-Slip Condition for a Mixture of Two Liquids

When a mixture of two viscous liquids flows past a solid wall there is an ambiguity in the use of the no-slip boundary condition. It is not obvious whether the mass-averaged velocity, the volume-averaged velocity, the individual species velocities, all or none of the above, or none of the above should exhibit no-slip. Extensive molecular dynamics simulations of the Poiseuille flow of mixtures of coexisting liquid species past an atomistic wall indicate that the velocity of each individual liquid species satisfies the no-slip condition and, therefore, so do mass and volume averages.

Quantification of the Performance of Agitators In Stirred Vessels: Definition and Use of an Agitation Index

An agitation index, based on velocity measurements obtained from laser Doppler velocimetry or other similar techniques, is proposed as an objective measure of the effectiveness of a particular agitator in inducing flow in a stirred vessel. This index is calculated by first generating a cell structure based on the measurement grid, and then assigning each velocity to the entire corresponding cell; the volume-weighted sum of velocities yields then the agitation index, which represents the volume-average velocity as a percentage of the impeller tip velocity. The usefulness of this objective measure of the quality of agitation in a stirred vessel is illustrated by applying it first to the case where three different agitators have to be graded in terms of effectiveness in inducing flow, and then to the case of determining the clearance of two agitators located on the same shaft corresponding to the optimum liquid circulation.

NMR flow velocity mapping in random percolation model objects: Evidence for a power-law dependence of the volume-averaged velocity on the probe-volume radius.

Two- or three-dimensional lacunar objects were fabricated using computer-simulated random sitepercolation networks as templates. The flow of water pumped through the pore space was studied with the aid of nuclear-magnetic-resonance- ~NMR! microscopy modified for mapping of the velocity vector field. The percolation backbones of the objects were determined by exclusion of all pixels or voxels of the spin-density images with velocities below the noise level. An evaluation procedure was established which reliably renders the fractal dimensions of the whole cluster and of its backbone. The volume-averaged velocity magnitude as a function of the probe-volume radius r was found to obey a power lawv V;r 2l in the range a,r,jv , where a is the voxel edge length and jv the velocity correlation length. The exponents turned out to be l50.3260.04 and l50.8260.03 for the two- and three-dimensional objects, respectively. In order to test the time dependence of the mean-squared displacement on fractals, ^r 2 &5at 2/dw, expected for random walks of a fractal dimension dw , self-diffusion of gaseous methane was examined in the pore spaces of the same objects with the aid of field-gradient NMR diffusometry. The results are in accordance with the theoretical predictions for anomalous diffusion on percolation clusters. This finding is supported by studies of incoherent water flow in the percolation network. @S1063-651X~96!06510-5#

Osmotic Dehydration: An Analysis of Fluxes and Shrinkage in Cellular Structure

A comprehensive analysis of fluxes and shrinkage in cellular structure immersed in osmotic solution was carried out. A relation between shrinking velocity and bulk flow velocity (volume average velocity) was developed for a shrinking body through volume change. Equations were derived to relate fluxes based on stationary frame and shrinking velocity frame, as well as based on the total interface area and true flow channel in cellular tissue. Computer simulations were carried out to show the trend and relative magnitude of diffusion, bulk flow and net fluxes of potato tissue slice immersed in 0.535 kmol/m3 mannitol solution. It was shown that the net volume flux ratio of water to mannitol was about 60 most of the time, but could be as high as 450 at the beginning of the process. In addition, the bulk flow transported about 90% of water removed in osmotic dehydration and washed back about 60% of mannitol penetrating by diffusion. Therefore the bulk flow should be considered in describing mass transfer in osmotic dehydration.

Numerical treatment of transient diffussion in shrinking or swelling solids

The diffusion equation for solids that undergo change of volume was numerically integrated, using a frame of reference fixed to the volume average velocity of the system. Numerical solutions are reported for absorption and desorption processes in swelling and shrinking systems, respectively, with slab, cylindrical or spherical geometry. Uniform initial concentration, constant surface concentration, symmetry with respect to the centre, central axis or central plane of the system and diffusion coefficient independent on diffusant concentration were assumed. Plots in terms of dimensionless concentration of diffusant versus Fourier number based on the initial half-thickness of the system are reported, covering different concentration ranges of diffusant.

On the peculiar velocity field of CDM universe

view Abstract Citations (6) References (25) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS On the Peculiar Velocity Field of a CDM Universe Brainerd, Tereasa G. ; Villumsen, Jens V. Abstract Using a large N-body simulation of a standard cold dark matter universe in which individual dark matter galaxy halos can be resolved, the peculiar velocity field of the halos and mass particles is investigated. The rms velocity (v_rms_), velocity correlation function [ξ_vv_(r)], and the three-dimensional pairwise velocity dispersion (σ_vv_) of the halos and mass are determined. The effects of halo mass and local environment on these functions are investigated. High-mass halos with overdensities of ~250 and ~70 are good tracers of the major mass motions in the simulation as defined by a volume-averaged velocity field. Low-mass halos are biased tracers of the same field. Assuming one luminous galaxy would reside in each of the halos and the mass-to-light ratio is a constant, this implies bright galaxies are fair tracers of the major mass motions in a CDM universe. The rms velocity of the halos is strongly affected by the local environment; the higher the background mass density the larger v_rms_. However, it is not straightforward to determine the magnitude of an enhancement/suppression in the local mass density given the local enhancement/suppression of v_rms_. At the end of the simulation, we obtain values of the velocity bias, b_v_ = σ_vv,h_/σ_vv,m_, as a function of halo mass and minimum overdensity similar to those found by Carlberg, Couchman, & Thomas and by Carlberg. For halos with overdensities of ~250 and ~70, b_v_ is a decreasing function of background mass density (i.e., the higher the background density, the larger the discrepancy between the velocity dispersions of the halos and mass particles), and for halos with overdensities of ~2000 it is an increasing function of background density (i.e., the lower the background density, the larger the discrepancy between the velocity dispersions of the halos and mass particles). The velocity bias as a function of scale, b_v_(r), is an increasing function of separation and even on large scales, r ~ 1400 km s^-1^, a noticeable velocity bias is present at the end of the simulation (~0.9 for halos with overdensities ~2000, ~0.8 for halos with overdensities ~250, and 0.7 for halos with overdensities ~70). Little difference is found between ξ_vv_(r) for halos and mass particles, and ξ_vv_(r) is similar for halos of high and low mass. Making the assumption that one luminous galaxy would reside in each of the halos and the mass-to-light ratio is a constant, this suggests the velocity correlation statistics of both faint and bright galaxies are representative of the velocity correlation statistics of the mass in a CDM universe. By the end of the simulation, ξ_vv_(r) is about 70% of the value predicted by linear theory over all scales investigated. Little environmental effect on ξ_vv_(r) of halos with overdensities of ~2000 is observed. On scales <~630 km s^-1^, the power in ξ_vv_(r) for halos with overdensities of ~200 and ~70 is due to halos in regions of high background mass density. Publication: The Astrophysical Journal Pub Date: December 1994 DOI: 10.1086/174926 Bibcode: 1994ApJ...436..528B Keywords: Astronomical Models; Cosmology; Dark Matter; Galactic Halos; Many Body Problem; Universe; Velocity Distribution; Computerized Simulation; Mathematical Models; Astrophysics; COSMOLOGY: DARK MATTER; COSMOLOGY: LARGE-SCALE STRUCTURE OF UNIVERSE; COSMOLOGY: THEORY; METHODS: NUMERICAL full text sources ADS |

Modeling of Cylindrical Alkaline Cells: VII . A Wound Cell Model

A one‐dimensional mathematical model of alkaline cells is used to simulate a wound geometry. It is the first simulation to describe the distribution of species, overpotential, solution current density, porosity, and volume average velocity within a wound cell to assess the reactant utilization. The model accounts for the porous nature of the electrodes and separator in a concentrated electrolyte at a high discharge rate. Virtual current collectors are introduced to transform the inherently two‐dimensional spiral geometry into a simplified one‐dimensional analysis. The results predict a superior material utilization for a wound cell rather than a bobbin design, due to a more uniform distribution of species created by an enhancement of the electrode‐separator interfacial area.

Fine-Sediment Deposition from Gravity Surges on Uniform Slopes

ABSTRACT The propagation of and the deposition from a noneroding, turbulent gravity surge are described by a simple model for a two-dimensional, well-mixed buoyant cloud of suspended particles moving down an inclined surface. The model includes the effects of entrainment of ambient seawater, deposition of suspended sediment, seafloor friction, and slope. Our results are applicable to large, decelerating turbidity currents and their distal deposits on uniform slopes in lakes and the sea. The scaling arguments that emerge from our analysis, moreover, have important ramifications for the design and interpretation of laboratory analogs of these phenomena. General solutions are obtained to the coupled equations that describe the evolution of momentum, total mass, and particulate mass of a surge. The solutions vary on two horizontal length scales: xo, beyond which the behavior of the surge is independent of the initial momentum and shape; and xr, beyond which the driving negative buoyancy of the surge is lost due to particle settling. For fine particles whose settling velocity is much less than the forward propagation speed of the surge, the suspension is well mixed and xo << xr, The deposit thickness diminishes as the inverse square root of the downstream distance x when xo << x << xr, and then diminishes exponentially with downstream distance as x approaches and exceeds xr. The length of a surge deposit scales with xr = kbosin/aws(cos)2, where k is the assumed constant aspect ratio of the surge, bo is the initial buoyancy per unit width at the point of issue onto a slope of constant angle ,a is the ambient density, ws is the average settling velocity of the suspended particles, and = 6 + 8CD/ incorporates the ratio of the constant coefficients of drag CD and fluid entrainment . Extension of our model to the case of two particle sizes indicates that, even for very poorly sorted suspensions, the estimate for the length of a surge deposit xr is valid if ws is defined as the volume-averaged settling velocity of the initial suspension at xo. The ratio of coarse to fine material in model deposits generated from initially poorly sorted suspensions can diminish dramatically in the downstream direction, however, due to differential rates of gravitational settling.

Direct measurement of interstitial velocity field variations in a porous medium using fluorescent-particle image velocimetry

Fluorescent-particle image velocimetry (FPIV) is used in conjunction with refractive-index matching to measure flow velocities in the interstitial regions of a porous medium. Adaptations that allow the use of conventional PIV methodology in index-matched systems are discussed, and the results of flow field measurements in a porous test bed under saturated flow conditions are presented. Preliminary analysis of these data, in which the horizontal and vertical components of the interstitial velocity vectors were averaged over a specified region within the medium, are also presented and compared with the vertical velocity component calculated from the total volumetric flow rate. The magnitude of the macroscopically derived and FPIV-measured values are found to be very similar. In general, our results demonstrate the utility of the FPIV technique to determine both the point velocity and the local volume-averaged velocity within porous media.

A macroscopic description of geometrical pore structure of porous solids

A description of macroscopic pore structure is discussed in regards to fluid flow through a porous media. Using volume and area averaging procedures for relative fluid flow, an anisotropic pore structure is defined by two parameters: the volume porosity and the second order structural permeability tensor. Such a characterization allows for a consideration of the fluid flow described by the volume average velocity as the superposition of three one-dimensional flows at the area average velocities along mutually perpendicular directions. This also leads to the understanding that only part of the fluid can flow unimpeded while the rest is trapped in the porous skeleton.

Flow in porous media I: A theoretical derivation of Darcy's law

Stokes flow through a rigid porous medium is analyzed in terms of the method of volume averaging. The traditional averaging procedure leads to an equation of motion and a continuity equation expressed in terms of the volume-averaged pressure and velocity. The equation of motion contains integrals involving spatial deviations of the pressure and velocity, the Brinkman correction, and other lower-order terms. The analysis clearly indicates why the Brinkman correction should not be used to accommodate ano slip condition at an interface between a porous medium and a bounding solid surface.

High Reynolds number flow through cubic arrays of spheres Steady-state solution and transition to turbulence

A five-mode truncation of the Navier—Stokes equation for incompressible flow through a cubic array of spheres is analyzed for its stationary and time-dependent solutions. Under the imposed symmetry requirements, the stationary solutions accurately depict the appearance of a vortex pair, and the volume-averaged velocity is in excellent agreement with the Ergun equation up to Re = 250. At the point the deviation from empirical data begins, the unique stationary solution destabilizes via a Hopf bifurcation point. A numerical analysis of the resulting periodic solution shows that for a certain range of imposed pressure gradient the system exhibits chaotic behavior, approached through a period-doubling subharmonic bifurcation sequence.