Fractal Models for Floc Density, Sedimentation Velocity, and Floc Volume Fraction for High Peclet Number Reactors

2015 ◽  
Vol 32 (12) ◽  
pp. 978-982 ◽  
Author(s):  
Monroe L. Weber-Shirk ◽  
Leonard W. Lion
Keyword(s):  
Peclet Number ◽  
Volume Fraction ◽  
Sedimentation Velocity ◽  
Péclet Number ◽  
Fractal Models

Magnetohydrodynamics and Soret Effects on Bioconvection in a Porous Medium Saturated With a Nanofluid Containing Gyrotactic Microorganisms

2014 ◽  
Vol 136 (5) ◽  
Author(s):  
S. Shaw ◽  
P. Sibanda ◽  
A. Sutradhar ◽  
P. V. S. N. Murthy
Keyword(s):  
Porous Medium ◽  
Differential Equations ◽  
Sherwood Number ◽  
Peclet Number ◽  
Volume Fraction ◽  
Soret Effect ◽  
Magnetic Field Effect ◽  
Lewis Number ◽  
Solute Concentration ◽  
Péclet Number

We investigate the bioconvection of gyrotactic microorganism near the boundary layer region of an inclined semi infinite permeable plate embedded in a porous medium filled with a water-based nanofluid containing motile microorganisms. The model for the nanofluid incorporates Brownian motion, thermophoresis, also Soret effect and magnetic field effect are considered in the study. The governing partial differential equations for momentum, heat, solute concentration, nanoparticle volume fraction, and microorganism conservation are reduced to a set of nonlinear ordinary differential equations using similarity transformations and solved numerically. The effects of the bioconvection parameters on the thermal, solutal, nanoparticle concentration, and the density of the micro-organisms are analyzed. A comparative analysis of our results with previously reported results in the literature is given. Some interesting phenomena are observed for the local Nusselt and Sherwood number. It is shown that the Péclet number and the bioconvection Rayleigh number highly influence the local Nusselt and Sherwood numbers. For Péclet numbers less than 1, the local Nusselt and Sherwood number increase with the bioconvection Lewis number. However, both the heat and mass transfer rates decrease with bioconvection Lewis number for higher values of the Péclet number.

On Simplified Models for the Rate- and Time-Dependent Performance of Stratified Thermal Storage

2007 ◽  
Vol 129 (3) ◽  
pp. 214-222 ◽  
Author(s):  
Ying Ji ◽  
K. O. Homan
Keyword(s):  
Entropy Generation ◽  
Peclet Number ◽  
Volume Fraction ◽  
Thermal Storage ◽  
Time Dependent ◽  
Simplified Models ◽  
Péclet Number ◽  
Thermal Mixing ◽  
Discharge Volume ◽  
Cumulative Entropy

In direct sensible thermal storage systems, both the energy discharging and charging processes are inherently time-dependent as well as rate-dependent. Simplified models which depict the characteristics of this transient process are therefore crucial to the sizing and rating of the storage devices. In this paper, existing models which represent three distinct classes of models for thermal storage behavior are recast into a common formulation and used to predict the variations of discharge volume fraction, thermal mixing factor, and entropy generation. For each of the models considered, the parametric dependence of key performance measures is shown to be expressible in terms of a Peclet number and a Froude number or temperature difference ratio. The thermal mixing factor for each of the models is reasonably well described by a power law fit with Fr2Pe for the convection-dominated portion of the operating range. For the uniform and nonuniform diffusivity models examined, there is shown to be a Peclet number which maximizes the discharge volume fraction. In addition, the cumulative entropy generation from the simplified models is compared with the ideally-stratified and the fully-mixed limits. Of the models considered, only the nonuniform diffusivity model exhibits an optimal Peclet number at which the cumulative entropy generation is minimized. For each of the other models examined, the cumulative entropy generation varies monotonically with Peclet number.

Mixing of the fluid phase in slowly sheared particle suspensions of cylinders

2017 ◽  
Vol 818 ◽  
pp. 807-837 ◽  
Author(s):  
Kjetil Thøgersen ◽  
Marcin Dabrowski
Keyword(s):  
Diffusion Coefficient ◽  
Fractal Dimension ◽  
Peclet Number ◽  
Volume Fraction ◽  
Fluid Phase ◽  
Particle Suspensions ◽  
Péclet Number ◽  
Self Diffusion ◽  
Wide Range ◽  
Self Diffusion Coefficient

We introduce a finite element model for neutrally buoyant particle suspensions of cylinders at zero Reynolds number and infinite Péclet number in the purely hydrodynamic limit, which allows us to access a high-accuracy fluid velocity field at any time during the simulation. We use the diffusive strip method to characterize the development of the concentration field in the fluid phase of sheared suspensions from initial thin filaments, and characterize the structures that form with their fractal dimension. We find that the growth of the fractal dimension of the filaments scales with the increase of mean square displacement in the fluid phase. Further, we measure the concentration distribution of tracers in the fluid phase, as well as the shear-induced self-diffusion coefficient in both the solid phase and the fluid phase. We demonstrate that the shear-induced self-diffusion coefficient is slightly larger in the fluid phase at infinite Péclet number. Finally, we investigate enhanced mass diffusivity in the fluid phase by systematic measurements of the shear-induced self-diffusion coefficient in the fluid phase for a wide range of fluid tracer Péclet numbers. We find that the functional dependence $D_{s}/D=1+\unicode[STIX]{x1D6FD}\unicode[STIX]{x1D719}^{\unicode[STIX]{x1D6FC}}Pe^{\unicode[STIX]{x1D701}}$ (where $D_{s}$ is the shear-induced self-diffusion coefficient, $D$ is the molecular diffusivity and $\unicode[STIX]{x1D719}$ is the particle volume fraction) fits the observations fairly well. We measure the constants $\unicode[STIX]{x1D6FD}=2.98\pm 0.39$, $\unicode[STIX]{x1D6FC}=1.61\pm 0.26$ and $\unicode[STIX]{x1D701}=0.900\pm 0.031$.

Dispersion in fixed beds

1985 ◽  
Vol 154 ◽  
pp. 399-427 ◽  
Author(s):  
Donald L. Koch ◽  
John F. Brady
Keyword(s):  
Peclet Number ◽  
Molecular Diffusion ◽  
Volume Fraction ◽  
Effective Diffusivity ◽  
Ensemble Averaging ◽  
Péclet Number ◽  
Physical Mechanisms ◽  
Molecular Diffusivity ◽  
Fixed Beds ◽  
Effective Diffusivities

A macroscopic equation of mass conservation is obtained by ensemble-averaging the basic conservation laws in a porous medium. In the long-time limit this ‘macro-transport’ equation takes the form of a macroscopic Fick's law with a constant effective diffusivity tensor. An asymptotic analysis in low volume fraction of the effective diffusivity in a bed of fixed spheres is carried out for all values of the Péclet number ℙ = Ua/Df, where U is the average velocity through the bed. a is the particle radius and Df is the molecular diffusivity of the solute in the fluid. Several physical mechanisms causing dispersion are revealed by this analysis. The stochastic velocity fluctuations induced in the fluid by the randomly positioned bed particles give rise to a convectively driven contribution to dispersion. At high Péclet numbers, this convective dispersion mechanism is purely mechanical, and the resulting effective diffusivities are independent of molecular diffusion and grow linearly with ℙ. The region of zero velocity in and near the bed particles gives rise to non-mechanical dispersion mechanisms that dominate the longitudinal diffusivity at very high Péclet numbers. One such mechanism involves the retention of the diffusing species in permeable particles, from which it can escape only by molecular diffusion, leading to a diffusion coefficient that grows as ℙ2. Even if the bed particles are impermeable, non-mechanical contributions that grow as ℙ ln ℙ and ℙ2 at high ℙ arise from a diffusive boundary layer near the solid surfaces and from regions of closed streamlines respectively. The results for the longitudinal and transverse effective diffusivities as functions of the Péclet number are summarized in tabular form in §6. Because the same physical mechanisms promote dispersion in dilute and dense fixed beds, the predicted Péclet-number dependences of the effective diffusivities are applicable to all porous media. The theoretical predictions are compared with experiments in densely packed beds of impermeable particles, and the agreement is shown to be remarkably good.

scholarly journals Single-particle motion in colloids: force-induced diffusion

2010 ◽  
Vol 658 ◽  
pp. 188-210 ◽  
Author(s):  
ROSEANNA N. ZIA ◽  
JOHN F. BRADY
Keyword(s):  
Peclet Number ◽  
Colloidal Particles ◽  
Volume Fraction ◽  
Spatial Configuration ◽  
Full Range ◽  
Thermal Fluctuations ◽  
Dynamics Simulation ◽  
Brownian Dynamics Simulation ◽  
Péclet Number ◽  
Long Time

We study the fluctuating motion of a Brownian-sized probe particle as it is dragged by a constant external force through a colloidal dispersion. In this nonlinear-microrheology problem, collisions between the probe and the background bath particles, in addition to thermal fluctuations of the solvent, drive a long-time diffusive spread of the probe's trajectory. The influence of the former is determined by the spatial configuration of the bath particles and the force with which the probe perturbs it. With no external forcing the probe and bath particles form an equilibrium microstructure that fluctuates thermally with the solvent. Probe motion through the dispersion distorts the microstructure; the character of this deformation, and hence its influence on the probe's motion, depends on the strength with which the probe is forced, Fext, compared to thermal forces, kT/b, defining a Péclet number, Pe = Fext/(kT/b), where kT is the thermal energy and b the bath particle size. It is shown that the long-time mean-square fluctuational motion of the probe is diffusive and the effective diffusivity of the forced probe is determined for the full range of Péclet number. At small Pe Brownian motion dominates and the diffusive behaviour of the probe characteristic of passive microrheology is recovered, but with an incremental flow-induced ‘microdiffusivity’ that scales as Dmicro ~ DaPe2φb, where φb is the volume fraction of bath particles and Da is the self-diffusivity of an isolated probe. At the other extreme of high Péclet number the fluctuational motion is still diffusive, and the diffusivity becomes primarily force induced, scaling as (Fext/η)φb, where η is the viscosity of the solvent. The force-induced microdiffusivity is anisotropic, with diffusion longitudinal to the direction of forcing larger in both limits compared to transverse diffusion, but more strongly so in the high-Pe limit. The diffusivity is computed for all Pe for a probe of size a in a bath of colloidal particles, all of size b, for arbitrary size ratio a/b, neglecting hydrodynamic interactions. The results are compared with the force-induced diffusion measured by Brownian dynamics simulation. The theory is also compared to the analogous shear-induced diffusion of macrorheology, as well as to experimental results for macroscopic falling-ball rheometry. The results of this analysis may also be applied to the diffusive motion of self-propelled particles.

Approximate analysis of the containment of contaminated sites prior to remediation

2000 ◽  
Vol 42 (1-2) ◽  
pp. 319-324 ◽  
Author(s):  
H. Rubin ◽  
A. Rabideau
Keyword(s):  
Steady State ◽  
Peclet Number ◽  
Dimensionless Parameter ◽  
Contaminated Sites ◽  
Unsteady State ◽  
Péclet Number ◽  
Vertical Barrier ◽  
Time Period ◽  
Barrier Performance ◽  
Vertical Barriers

This study presents an approximate analytical model, which can be useful for the prediction and requirement of vertical barrier efficiencies. A previous study by the authors has indicated that a single dimensionless parameter determines the performance of a vertical barrier. This parameter is termed the barrier Peclet number. The evaluation of barrier performance concerns operation under steady state conditions, as well as estimates of unsteady state conditions and calculation of the time period requires arriving at steady state conditions. This study refers to high values of the barrier Peclet number. The modeling approach refers to the development of several types of boundary layers. Comparisons were made between simulation results of the present study and some analytical and numerical results. These comparisons indicate that the models developed in this study could be useful in the design and prediction of the performance of vertical barriers operating under conditions of high values of the barrier Peclet number.

Convective diffusion toward the sphere in laminar flow around the sphere

1979 ◽  
Vol 44 (4) ◽  
pp. 1218-1238
Author(s):  
Arnošt Kimla ◽  
Jiří Míčka
Keyword(s):  
Laminar Flow ◽  
Velocity Profile ◽  
Peclet Number ◽  
Convective Diffusion ◽  
Péclet Number ◽  
Stability Of The Solution

The problem of convective diffusion toward the sphere in laminar flow around the sphere is solved by combination of the analytical and net methods for the region of Peclet number λ ≥ 1. The problem was also studied for very small values λ. Stability of the solution has been proved in relation to changes of the velocity profile.

Convective diffusion to a slowly rotating spherical electrode; Basic model for Re → O, Pe → ∞

1983 ◽  
Vol 48 (6) ◽  
pp. 1571-1578 ◽  
Author(s):  
Ondřej Wein
Keyword(s):  
Boundary Layer ◽  
Flow Regime ◽  
Peclet Number ◽  
Convective Diffusion ◽  
Creeping Flow ◽  
Péclet Number ◽  
Spherical Electrode ◽  
Basic Model ◽  
Boundary Layer Solution ◽  
Layer Solution

Theory has been formulated of a convective rotating spherical electrode in the creeping flow regime (Re → 0). The currently available boundary layer solution for Pe → ∞ has been confronted with an improved similarity description applicable in the whole range of the Peclet number.

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