BBO Equation Research Articles

Feasibility Study of Drill Cuttings Slip Velocity Estimation using BBO DTM-Pade’ Semi-analytical Technique

The study reviewed the use of DTM-Pade’ approximation method of Basset–Boussinesq–Oseen equation (BBO) in predicting particle slip velocity. A hypothetical example is used in the study with three different diameters, 3,5 and 7mm, of spherical particle assumed to represent wide rang of drill cuttings’ sizes. In addition, the drilling fluid is assumed to be incompressible fluid in stationary mode. The Pade’ approximation (K=20, [8 8]) is used, based on literature, in calculating slip velocities through 2 seconds time span. It is concluded that this technique is a practical mean of particle slip velocity prediction at high Pad’ orders using Matlab for a short period of time, around (o.1 to 0.2 seconds). However, BBO equation application range such as, single particle fall in infinite incompressible fluid, small particle size rang (less than 3mm), low particle to fluid density ratio (higher than 0.38), limits its application in oil well drilling operations. Moreover, the necessity of more factors considerations in simulating drilling operations including Non-Newtonian fluids, wall effect, and wide range of irregular shaped drill cuttings densities and sizes urges for the use of generalized BBO equation. In drilling operations, concisely, drill cuttings are usually mixed in the wellbore annulus and for more reliable predictions the application of more sophisticated techniques such as Machine Learning (MA) and Artificial Intelligence (AI) become indispensable.

General Basset–Boussinesq–Oseen equation: existence, uniqueness, approximation and regularity of solutions

The Basset–Boussinesq–Oseen equation (BBO equation) is the mathematical formulation of the Lagrangian acceleration of a spherical particle, moving with Lagragian velocity, in an unsteady flow as the sum of the viscous, gravitational, buoyancy, virtual mass, and Basset forces acting respectively on the particle. Despite the widespread use of the equation in applications, the basic properties of its solutions have remained unexplored. Here we fill this gap by proving global existence and uniqueness of solutions in an appropriate partially ordered Banach space. The method also gives an approximate solution to the problem with accurate result which is easy to implement. Meanwhile, regularity properties of the solutions are proved under some conditions which gives a representation of the solutions. Illustrative examples exhibit the efficiency of our method.

Motion of tracer particles in a centrifugal pump and its tracking characteristics

The Basset-Boussinesq-Oseen (BBO) equation can be used for most flows to trace the motion of a particle, but in a centrifugal pump, among the forces that act on the particles, one should also include those due to the impeller rotation, as additional effects. This paper firstly reviews various approximations of the BBO equation for the motion of dispersion particles in a viscous fluid. Then based on the motion equation for particles in low Reynolds number centrifugal pumps, a formula for calculating the tracking characteristics of tracer particles is deduced through the Fourier integral transformation. After that the deviations of the particle motion from the fluid motion, as predicted by the various approximations, are discussed and compared. At last, with an emphasis on the Particle Image Velocimetry (PIV) results, the tracking characteristics of particles are estimated. Also, advantages and disadvantages of different tracer particles are discussed and suitable tracer particles for application in PIV studies for flow fields in centrifugal pumps are suggested.

Numerical simulation of growing Cu particles in a Kenics static mixer reactor in which Cu 2+ is reduced by carbohydrates

Numerical simulations of the mixing of copper ions (Cu(II)) and growing particles have been performed in the range of Re 10–1000 in a vertical (down-flow) Kenics static mixer reactor. The growth of the particles is based on the local Cu(II) concentration and determines the reduction rate of the Cu(II) concentration, which is solved with a finite volume code. The particles are tracked using a code that solves the BBO equation and the flow is solved using a Lattice Boltzmann code. It was found that the Kenics static mixer enhances the mixing of Cu(II). However, at low Reynolds numbers ( Re < 500 ) the particles are not well mixed, which has a negative influence on the particle size distribution (PSD). Furthermore, in a pipe (without a mixer) the PSD is more narrow than in a Kenics static mixer reactor, but the Cu(II) concentration is not well mixed in a pipe, which slows down the reduction process. Results indicate that a high Reynolds number is necessary to mix both the particles and Cu(II) concentration in the Kenics static mixer.

Numerical Investigation of Behaviour of Small Particle Passing through Normal Shock Wave (2nd Report, Behaviour of Small Particle Passing through Oscillating Shock Wave and Error in Measurement)

An anemometry using tracer particle, such as PIV and LDV, has some errors related to the deviation of particles from the stream, particularly around the large velocity gradient. But the reasonable correcting procedure has never been established. For a notable example, we have investigated the behavior of particle when it passes through a shock wave. In the first report, the effects of particle density and size are clarified by carrying out the simulation with the weak shock theory to the flow field and the BBO equation to the traveling particle. In the present paper, shock wave oscillation and nonuniforme particle size are taken into account, to realize the more exact experimental conditions. The results suggest that the turbulence detected in measurement contains the considerably large amount of errors concerning with the particle property used. In particular, the dispersion of particle diameter is more serious.

A Numerical Investigation of Particle Deposition on a Square Cylinder Placed in a Channel Flow

Particle deposition on a square cylinder placed in a rectangular channel was investigated for unsteady vortical flows. For the two-phase flow simulations, the unsteady gas flow field was computed by solving the incompressible Navier-Stokes equations using a staggered-grid control volume approach and the Marker-and-Cell (MAC) technique. The particle dynamics were simulated using the modified Basset-Bousinesq-Oseen (BBO) equation. The gas-phase algorithm was validated using four test problems involving both steady and unsteady flows. Numerical experiments were also conducted to evaluate the relative contributions of various terms in the BBO equation. For particle dynamics in unsteady vortical flows, all the secondary terms were found to be negligible compared to the steady state viscous term at particle density ratios >20. The two-phase flow model and the detailed flow visualization were then employed to characterize particle dispersion and deposition as a function of the Reynolds number, particle Stokes number (St), and density ratio (e). Particle dispersion in the cylinder wake exhibited a typical nonmonotonic behavior. Particles with St 1.0 were essentially unaffected by the flow in the near wake region. In addition, the small St particles were distributed in the vortex core, while the intermediate St particles were distributed around the vortex periphery. For e > 20, the particle deposition was essentially characterized by the Stokes number. The amount of deposition increased precipitously as St was increased from zero to unity, then increased slowly for St between 1 to 3, and was essentially independent of St for St > 3.0. For the range of Reynolds numbers investigated, which included both laminar and transitional regimes, the Reynolds number (Re) had a negligible effect on particle deposition, but a more discernible effect on particle distribution and dispersion.

A Review of Droplet Dynamics and Vaporization Modeling for Engineering Calculations

The present paper reviews the methodologies for representing the droplet motion and vaporization history in two-phase flow computations. The focus is on the use of droplet models that are realistic in terms of their efficient implementation in comprehensive spray simulations, representation of important physical processes, and applicability under a broad range of conditions. The methodologies available at present to simulate droplet motion in complex two-phase flows may be broadly classified into two categories. First one is based on the modified BBO equation. This approach is more comprehensive, but requires modifications and/or correlations at higher droplet Reynolds number. The second approach aims at developing correlations, using detailed numerical simulations or laboratory experiments, for the effects of flow nonuniformity and droplet relative acceleration on the instantaneous drag and lift coefficients. Recent advances made in the droplet vaporization models are also discussed. The advanced vaporization models include the effects of transient liquid heating, gas-phase convection, and variable thermo physical properties. All of these models are discussed, and recommendations are made for their inclusion in comprehensive two-phase computations.

Motions of Small Rigid Spheres in Simulated Random Velocity Field

Motions of particles in a simulated three‐dimensional random Gaussian flow field are numerically studied. The equation of motion of a solid particle (the generalized BBO equation), in addition to the Stokes drag, the Basset force, and the virtual mass effects, includes the three‐dimensional lift force. An ensemble of 300 particle trajectories are generated for evaluating various statistics. Turbulent diffusivity of small particles is calculated and its variations with nondimensional particle size, particle Reynolds number, density ratio, and particle response time are studied. Particle velocity autocorrelations and particle velocity‐fluid velocity cross‐correlations, as well as fluctuation energy transport from the fluid phase to the particulate phase, are evaluated. A simplified model for evaluating the particle response statistics is also described. The simulation results are compared with the predictions of the simplified model and the available experimental data and good agreement is observed. Effects...

Comment on "A Prediction of Particle Behavior via the Basset-Boussinesq-Oseen Equation"

N a recent Note, Catalano1 has given an analytical solution of the Basset-Boussinesq-Oseen (BBO) equation. This solution was applied to the interesting problem of the discrepancy between theory and laser velocimetry applications (LDA) measurements in the case of unstationary flows arising especially in Raleigh-Benard convection or at the onset of turbulence in a circular Couette flow. This discrepancy is observed mainly for high frequencies. In the following we would like to comment on the author's assumptions: Accepting, as Catalano did, the BBO equation in the form given by Tchen2 (for a thorough discussion of this topic, we refer to Reek's recent paper),5 we do not agree that the Basset term can be neglected. For example, we look at the experiments cited by Catalano6'7: Taking into account the properties of liquid helium, it turns out that Hinze's4 conditions for the neglect of the Basset term are not fulfilled for marker particles with a diameter less 10 \im. Also, it should be mentioned that Catalano neglected the mass addition term, which represents the force to accelerate the virtual mass of the particle relative to the fluid. In extending Catalano's analytical investigation, we present a solution of the complete BBO equation. We use a standard method3 for solving the inhomogeneous Volterraintegral equation of second kind. Our solution accounts for the Basset term describing the time behavior of the flow in the neighborhood of the particles. We proceeded in this way because we believe that, in unstationary flows of the type considered, near the onset of turbulence, the Bassset term cannot be neglected. The resultant closed-form solution for the particle motion should be examined for the case of Rayleigh-Benard flow as well. The nomenclature we use is the same as Catalano's. At this opportunity we also want to make some corrections of Eqs. (2), (3), (6), (7), (15), (17), (18), (20), and (21) of Ref. 1. Further, we would like to mention that Catalano did not remark that the lift forces [see Eqs. (4) and (5)] are directed perpendicular to the relative velocity vector up—uF. So the inclusion of the lift force is rather questionable in the obviously one-dimensional case presented.1