Mass transfer between bubbles and the continuous phase in a fluidized bed of variable cross section

A study is made of mass transfer in an inhomogeneous fluidized bed whose cross section varies over the height. The field of the liquid phase around a bubble is constructed, conditions are obtained for the existence of a region of closed circulation of the liquid phase, and the boundaries of this region are determined. A solution is given to the problem of convective diffusion of the substance to the region of closed circulation, and the mass-transfer coefficient between a bubble and the continuous phase is determined as a function of the flow parameters. By the same token, the results of [1] are generalized to a fluidized bed of variable cross section. It is shown that in this case the mass transfer is improved.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 20–29, November–December, 1974.     

Approximate solutions for power-law fluid flow past a particle at low Reynolds numbers

A simple perturbation approximation is proposed for describing flow behaviour of particles immersed in a uniform flow and an extensional flow of power-law fluids. The present solution for particles in a uniform flow field is in good agreement with the numerical solution in the literature. Theoretical predictions indicate that the effect of pseudoplasticity on flow around particles in an extensional flow field is small compared with that for particles in a uniform flow field.From the viewpoint of perturbation techniques, existing analytical solutions based on linearization of the equations of motion for particle in a power-law fluid are re-examined. Mass transfer to a power-law fluid from a particle is also discussed.     

Motion of bubbles in a fluidized bed

The steady-state motion of a bubble (a cavity free from suspended particles and occupied solely by the liquid phase) in a fluidized bed of uniform concentration is considered. The change in the shape of the bubble which takes place as it rises through the fluidized bed is established; the rising velocity is determined for both large and small bubbles. The basic parameter characterizing the shape of a large bubble in either a fluidized bed or a homogeneous liquid is calculated. This, in particular, enables the well-known Taylor problem of a large drop or bubble in an unlimited medium to be solved.     

Transverse galloping at low Reynolds numbers

The possibility of transverse galloping of a square cylinder at low Reynolds numbers (Re≤200, so that the flow is presumably laminar) is analysed. Transverse galloping is here considered as a one-degree-of-freedom oscillator subjected to fluid forces, which are described by using the quasi-steady hypothesis (time-averaged data are extracted from previous numerical simulations). Approximate solutions are obtained by means of the method of Krylov-Bogoliubov, with two major conclusions: (i) a square cylinder cannot gallop below a Reynolds number of 159 and (ii) in the range 159≤Re≤200 the response exhibits no hysteresis.     

On the breakup of bubbles at high Reynolds numbers and subcritical Weber numbers

In this paper we propose a simplified two-dimensional model to describe some aspects of the turbulent breakage of bubbles at subcritical Weber numbers. In particular we focus on the breakup of bubbles owing to their interaction with an array of successive eddies, modeled by a train of straining flows. Our simulations show that, under certain conditions, a bubble accumulates energy due to its interaction with a sequence of turbulent structures until it eventually breaks, even if none of the eddies is sufficiently energetic to split the particle by itself. It is also shown that the different strain directions of the eddies acting on the surface of the bubble, and the resonance effect between their characteristic frequency and the natural oscillation frequency of the bubble immersed into the straining flow are the two key factors in the bubble deformation, and subsequent breakup mechanism. Moreover, the breakup patterns obtained from our simulations seem to agree qualitatively well with the experimental observations.     

Diffusion to a particle at large péclet numbers and small Reynolds and Hartmann numbers

A study is made of the influence of a homogeneous magnetic field on the mass transfer for a spherical solid particle and a liquid drop in a flow of a viscous electrically conducting fluid. The previously obtained [1] velocity field of the fluid is used to calculate the concentration distribution in the diffusion boundary layer, the density of the diffusion flux, and the Nusselt number, which characterizes the mass transfer between the particle and the surrounding medium.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 189–192, January–February, 1980.     

Flow past a square cylinder at low Reynolds numbers

Results are presented for the flow past a stationary square cylinder at zero incidence for Reynolds number, Re ? 150. A stabilized finite‐element formulation is employed to discretize the equations of incompressible fluid flow in two‐dimensions. For the first time, values of the laminar separation Reynolds number, Re_s, and separation angle, θ_s, at Re_s are predicted. Also, the variation of θ_s with Re is presented. It is found that the steady separation initiates at Re = 1.15. Contrary to the popular belief that separation originates at the rear sharp corners, it is found to originate from the base point, i.e. θ_s=180^° at Re = Re_s. For Re > 5, θ_s approaches the limit of 135 ^°. The length of the separation bubble increases approximately linearly with increasing Re. The drag coefficient varies as Re^?0.66. Flow characteristics at Re ? 40 are also presented for elliptical cylinders of aspect ratios 0.2, 0.5, 0.8 and 1 (circle) having the same characteristic dimension as the square and major axis oriented normal to the free‐stream. Compared with a circular cylinder, the flow separates at a much lower Re from a square cylinder leading to the formation of a bigger wake (larger bubble length and width). Consequently, at a given Re, the drag on a square cylinder is more than the drag of a circular cylinder. This suggests that a cylinder with square section is more bluff than the one with circular section. Among all the cylinder shapes studied, the square cylinder with sharp corners generates the largest amount of drag. Copyright © 2010 John Wiley & Sons, Ltd.     

Free and confined jets at low Reynolds numbers

Free jets, and jets with tubular confinements, are investigated in the jet Reynolds number regime 80 Re_j 1000 being of interest for micro-jet pumps, among other applications. For issuing the jets, conventional (single-hole) nozzles as well as dual-hole nozzles of a particular design are used. Both flow visualization and LDA measurement indicate that, in agreement with previous findings, the jets issuing from conventional nozzles remain laminar up to large distances from the orifice. Thus there is but little entrainment of ambient fluid, and the performance of conventional nozzles in micro-jet pumps is rather poor. The dual-hole nozzles, on the other hand, are found to enforce transition to turbulent flow near the orifices. As a result, the entrainment rate is considerably increased, and the performance of jet pumps is improved when the dual-hole nozzles are applied. The experimental data are found to be in fair agreement with predictions based on mass and momentum balances.     

Mass transfer between bubbles and the dense phase in gas fluidized beds

Mass transfer between a bubble and the dense phase in gas fluidized beds of Group A and Group B particles was proposed based on previous experimental results and literature data.The mass transfer coefficient between bubbles and the dense phase was determined by k_(be) = 0.21d_b.A theoretical analysis of the mass transfer coefficient between a bubble and the dense phase using diffusion equations showed that the mass transfer coefficient between a bubble and the dense phase is k_(be) ∝ε_(mf)(Du_b/d_b)~(1/2) in both three- and two-dimensional fluidized beds.An effective diffusion coefficient in gas fluidized beds was introduced and correlated with bubble size as De = 13.3d_b~(2.7)7 for Group A and Group B particles.The mass transfer coefficient k_(be) can then be expressed as k_(be) = 0.492ε_(mf)(u_bd_b~(1.7))~(1/2) for bubbles in a three-dimensional bed and k_(be) = 0.576ε_(mf)(u_bd_b~(1.7))~(1/2) for bubbles in a two-dimensional bed.     

A model of bubbles rising in a fluidized bed

A model for a single fully developed bubble moving in an unbounded fluidized bed is presented. The model allows bubble growth or shrinkage during the rise inside the bed, as well as dependence of the rise velocity upon specified bed parameters. Limiting cases of nearly spherical bubbles and of sufficiently large bubbles whose form resembles that of a spherical segment are considered in more detail. The form of bubbles rising in either fluidized beds or one-phase liquids, and its dependence on the effective “surface tension” acting on the bubble boundary are discussed.     

Forced convection heat transfer from an equilateral triangular cylinder at low Reynolds numbers

An unsteady two-dimensional numerical simulation is performed to investigate the forced convection heat transfer for flow past a long heated equilateral triangular cylinder in an unconfined medium for the low Reynolds number laminar regime. The Reynolds number considered in this study ranges from 50 to 250 with three different values of Prandtl number (Pr?=?0.71, 7 and 100). Fictitious confining boundaries are chosen on the lateral sides of the computational domain that makes the blockage ratio β?=?5?% in order to make the problem computationally feasible. An unstructured triangular mesh is used for the computational domain discretization and the simulation is carried out with the commercial CFD solver Fluent. The flow and heat transfer characteristics are analyzed with the streamline and isotherm patterns at various Reynolds numbers. The dimensionless frequency of vortex shedding (Strouhal number), drag coefficient and Nusselt numbers are presented and discussed. The results obtained are in good agreement with the available results in the literature.     

Forces on a spherical particle in shear flow at intermediate Reynolds numbers

When particles are submerged in a shear flow, there are lateral (lift) forces on the particles, and these lateral forces affect the dispersion of the particles very much. Recent literature survey indicates that there are large discrepancies among the results from the previous numerical investigations on this subject. A small computational domain ranging between 20–30 sphere radii was used in all the previous numerical investigations. However, the result from the present study reveals that the value of lift coefficient strongly depends on the size of computational domain. To provide correct numerical data and physical interpretation for the forces on a spherical particle in linear shear flow, accurate numerical computations were performed for 5≤Re≤200 using a computational domain of 101 sphere radii.     

Magnetohydrodynamic flow past a drop at low Reynolds and Hartmann numbers

The flow of an electrically conductive liquid past a solid spherical particle at low Reynolds and Hartmann numbers in longitudinal and transverse magnetic fields was first investigated in [1,2]. The effect of a weak magnetic field on the strength of the resistance of a conductive drop in a dielectric medium was considered in [3]. In the present paper we consider the motion of a conductive liquid drop in an electrically conductive medium and calculate the strength of the resistance in the Stokes approximation for an arbitrary orientation of the uniform magnetic field and in the Oseen approximation for the case in which the direction of the magnetic field coincides with the direction of the oncoming stream. As in the previous studies, we do not consider the possibility of the formation of a double layer on the interface between the phases.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 19–25, November–December, 1978.The authors are grateful to G. I. Petrov and the participants in the seminar they conducted for their comments on the work.