Modified Reynolds Number Research Articles

Transport phenomena in an agitated vessel with an eccentrically located impeller

AbstractThe paper presents results of an experimental analysis of the transport phenomena at the vicinity of the wall of an unbaffled agitated vessel with an eccentrically located impeller. Distributions of the transport coefficients were experimentally studied using an electrochemical method within the turbulent regime of the Newtonian liquid flow. Measurements were carried out in an agitated vessel with the inner diameter T = 0.3 m. Liquid height in the vessel was equal to the inner diameter, H = T. The agitated vessel was equipped with a Rushton or a Smith turbine or an A 315 impeller. Eccentricity of the impeller shaft was varied from 0 to 0.53. Local values of the dimensionless shear rate, shear stress, dynamic velocity and friction coefficient were integrated numerically for the whole surface area of the cylindrical wall of the vessel. Averaged values of these quantities were correlated with the impeller eccentricity and modified Reynolds number. The proposed Eqs. (5)–(8), with the coefficients given in Table 2, have no equivalent in open literature concerning this subject. Distributions of the shear rate, γ/n, and friction coefficient, f, at the vicinity of the cylindrical wall of the unbaffled vessel equipped with eccentric Rushton or Smith turbine or A 315 impeller are very uneven and they depend significantly on the impeller eccentricity, e/R. Maximum local values of these variables are located on the wall section closest to the impeller blades. From among the tested impellers, the greatest effects of the impeller eccentricity, e/R, and the liquid turbulence (described by the modified Reynolds number Re P,M) on the averaged dimensionless shear rate (γ/n)m and friction coefficient, f m, are found for the radial-flow Rushton turbine located eccentrically in an unbaffled agitated vessel.

Forced flow and convective melting heat transfer of clathrate hydrate slurry in tubes

In this study, we presented the forced flow and convective melting heat transfer characteristics of tetrabutylammonium bromide (TBAB) clathrate hydrate slurry (CHS) flowing through straight circular tubes. Pressure drops of TBAB CHS with the volume fraction in the range of 0–20.0 vol% were measured, and the influence of the volume fraction was analyzed. Power-law equation was used to describe the flow behavior of CHS, and the corresponding indicators including flow behavior index and fluid consistency coefficient indicated that TBAB CHS behaved as a pseudo plastic fluid. Based on the modified Reynolds number, it was found that the empirical correlation could be used to predict the flow friction factor of TBAB CHS. Heat transfer experiments were conducted to measure the local heat transfer coefficients of TBAB CHS flow under constant wall heat flux, which were found to be enhanced by the latent heat involved in the solid–liquid phase change. Local heat transfer coefficient decreased along the heating section until the volume fraction was smaller than a certain small value, then it increased in the following section due to the increase of Reynolds number. The effects of heating power, flow velocity, volume fraction on the local heat transfer coefficients were discussed. Correlations for predicting TBAB CHS heat transfer performance were proposed based on the experimental data in laminar, transitional and turbulent flow regions, respectively.

Mass Transfer from Suspended Sphere in Presence of Inert Multi-Particulate Fluidised Solid

Mass transfer from naphthalene spheres in the presence of inert fluidised particles of different size fraction has been studied. The parameters considered in the present study include effect of particle Reynolds number Re p , suspended sphere diameter, bed porosity in mass transfer, and effect of inert fluidised particle diameter on mass transfer coefficient. Inert fluidised particles, spherical in nature and having specific gravity 1.4 with size ranging from 0.2923667 to 0.6276975 mm were used. The diameter of the suspended naphthalene spheres varied from 17 to 31 mm. The Re p varied from 11 to 70 with varying air velocity and fluidised bed porosity varied from 0.48 to 0.73. Mass transfer coefficient was found to vary from 3.6 ¥ 10− 5 to 2.4 ¥ 10− 4. The corresponding mass transfer factor varied from 7.88 ¥ 10− 7 to 8.92 ¥ 10− 6. The effect of different parameters, which affect the rate of mass transfer from suspended spheres in the presence of inert fluidised particles, were co-related in a new form of mass transfer factor J D ′ which is the function of mass transfer factor J D , naphthalene diameter D nap and inert particle diameter D pi . A new correlation was established with J D ′ and modified Reynolds number.

HEAT TRANSFEROFNON-NEWTONIAN FLUID FLOW INA CHANNEL LINED WITH POROUS LAYERS UNDER THERMAL NONEQUILIBRIUM CONDITIONS

Forced convection of a non-Newtonian fluid in a channel partially filled with porous layers is studied. In order to study the problem in its most realistic conditions, the Brinkman-Forchheimer model for momentum conservation and two equations, the local thermal equilibrium and nonequilibrium models for energy conservation, are used. Flow is assumed to be fully developed, but development of the thermal boundary layer is taken into account. The effect of the power-law index of the non-Newtonian fluid (n) as well as thermal conductivity of the solid matrix and the thickness of the porous layer are studied. It is shown that thinner porous layer fluids with smaller n show better heat-transfer capability in the same modified Reynolds number, whereas when the porous layer fluids are thicker, fluids with larger n have a greater Nusselt number, except when the thickness ratio is very close to 1.

Damping and added mass coefficients for a squeeze film damper using the full 3-D Navier–Stokes equation

Direct and cross-coupled damping coefficients are developed for the 2π-film, π-film (Gumbel cavitation condition) and homogeneous two-phase mixture films in a squeeze film damper. The numerical simulation uses the CFD-ACE+ commercial software, which employs a finite volume method for the discretization of the Navier–Stokes equations (NSE). In order to determine the dynamic coefficients, the NSE is combined with a finite perturbation method applied to the ‘equivalent journal’ of the damper. It was found that for the 2π-film and the Gumbel conditions, the damping coefficients exhibit linear characteristics, while the homogeneous cavitation model yields nonlinear coefficients. Using the CFD-ACE+, the inertia/added mass coefficients are derived for the limiting cases of the short and long dampers, respectively. The first set of forces is calculated by setting the fluid density to its actual value. The second set of forces is calculated when the density of the fluid is set close to zero (1E-10 kg/m 3), thus practically eliminating the effects of the inertia terms. Subtracting the two sets of forces from each other, allows the determination of the inertia component contribution and the corresponding inertia coefficients. By varying the density, dynamic viscosity and whirling speed, it was found that the inertia coefficients follow a single curve represented by a function dependent on the modified Reynolds number, Re*. The inertia coefficients presented in this study are compared with the ones reported by other researchers that used the modified Reynolds equation. Some differences were found between the NSE based results and the Reynolds equation based outcomes. This is attributed to the three-dimensional effects introduced by the totality of the terms comprised in the full NSE.

Heat transfer model of single-phase natural circulation flow under a rolling motion condition

Experimental studies on heat transfer characteristics for single-phase natural circulation flow under a rolling motion condition are performed. Experiments with and without rolling motions are conducted so that the effects of rolling motion on natural circulation heat transfer are obtained. The experimental results show rolling motion enhances the heat transfer. The heat transfer coefficient of natural circulation flow increases with the rolling amplitude and frequency. A modified Reynolds number that considers the influence of the acceleration is employed to express the effect of heat capacity. Using experimental data, an empirical equation for the heat transfer coefficient under a rolling motion condition is obtained. The calculated results agree with the experimental data.

Planar sudden symmetric expansion flows and bifurcation phenomena of purely viscous shear-thinning fluids

The goal of a present study is to investigate the effects of the generalized Newtonian fluids on the threshold of the transition from flow symmetry to its asymmetry for the flow through a 1:3 planar sudden expansion. We consider purely viscous shear-thinning fluids covering a range of the index n of Power law model n = 0.6 , 0.8 and compare them with the Newtonian fluid ( n = 1.0 ). Sudden expansion fluid flow is studied numerically by solving the two-dimensional momentum equations along with the continuity equation and the Power law rheological model. We report a systematic results in a range of generalized Reynolds number 10 ≤ R e g e n ≤ 150 with a focus on its critical value. The subsequent effects are investigated by means of a different flow parameters, i.e. generalized Reynolds number ( R e g e n ), modified Reynolds number ( R e m o d ) and the Reynolds number defined in accordance with the inlet wall viscosity ( R e w a l l ). Results indicate that the shear-thinning viscous behaviour increases the onset of bifurcation phenomena and the critical value of Reynolds number.

Analyzing threshold velocities for fluidization and pneumatic conveying

In previous works we have shown that by defining various threshold velocities in terms of the modified Reynolds number as a function of the modified Archimedes number a generalized master-curve can be plotted. Then, using this curve, the ratio between various threshold velocities can be easily determined. However, all the threshold velocities were defined for mono-sized particles. The purpose of this paper is to show how the threshold velocities can be used to design of particle–fluid systems for particle size distribution. The analysis provides practical guides for various engineering scenarios, such as selecting the appropriate fluidization velocity for maximum fluidization and minimum entrainment and determining the conveying velocity in pneumatic systems. In addition, the analysis provides a guide to determine whether the deposited layer of particles at the pipe bottom is stationary or moving, for cases where the superficial velocity is smaller than the saltation velocity.

Boundary saltation and minimum pressure velocities in particle–gas systems

The saltation velocity is one of the key design parameters in pneumatic conveying systems. The aim of this work is to experimentally study the mechanism of saltation. Experiments were carried out with various spherical and non-spherical particles in a small wind tunnel with very dilute flow. For each velocity the distribution of halted particles along the tunnel bottom was measured. From this length distribution, the median length was determined and used for further analysis. It was found that the median conveying length approaches infinity at a certain threshold velocity. By testing many materials the boundary saltation velocity followed a simple correlation of the modified Reynolds number as a function of the modified Archimedes number. The conveying length was an accumulation of three lengths: the first flight length, the rebound length, which is affected by the coefficient of restitution, and the rolling/sliding length, which is affected by the coefficient of friction. By analyzing these lengths, the total conveying length and the boundary saltation velocity were easily defined. Furthermore, as the velocities at minimum pressure velocities (referred to in the paper as the minimum pressure velocities) have been found to follow the same trend as the boundary saltation velocity if the solid concentration is taken into account, our simple correlation can describe by ± 30% all the relevant experiments found in the literature.

Production of nanosized powders of ferroelectrics with narrow particle size distributions

217 Functional materials containing ferroelectric crystals of metatitanates of divalent metals (barium, strontium, lead) are promising for manufacturing optical and piezo ceramics, ceramic memory elements, and artificial photon structures with energy gaps [1‐6]. To obtain such materials, it is necessary to have single-phase ferroelectrics with stoichiometric composition in the form of nano- and microcrystalline powders with given narrow particle size distributions. Methods for producing powders with a given particle size have not yet been developed. In this work, by the example of metatitanates of divalent metals, we propose a new approach to controlling the sizes of powders of ferroelectric materials and describe the use of this approach for producing nanosized powders with given narrow particle size distributions. A suspension containing powders of metatitanates of divalent metals (barium, strontium, and lead) synthesized according to a published procedure [7] was subjected to hydrodynamic treatment. The stirring intensity was estimated from the modified Reynolds number [8]

Large-eddy simulations of the turbulent Hartmann flow close to the transitional regime

A series of large-eddy simulations (LES) of turbulent and transitional channel flows of a conductive fluid under the effect of a uniform magnetic field applied in the wall-normal direction, usually referred to as Hartmann flows, are performed using the dynamic Smagorinsky model. The flow is characterized by the hydrodynamic Reynolds number Re and the Hartmann number Ha that is related to the strength of the external magnetic field. Previous measurements and works on stability analysis have shown that a critical modified Reynolds number, based on the Hartmann layer thickness R=Re∕Ha, exists at approximately R=380. Here, the LES are used to investigate the laminarization of flow when decreasing R. Also, similarities of the turbulent flows are explored for different values of Re and Ha but the same values of R or of the interaction parameter N=Ha2∕Re. The LES confirm that R is the relevant parameter that describes the transition and a critical Reynolds number for relaminarization is observed at R≈500. However, for turbulent flows at moderate values of the Ha and Re as considered in this study, the similarity with respect to N appears to be better than for R, especially for the first order statistics. Finally, the importance of the dynamic Smagorinsky model as the Hartmann number increases is discussed.

Mixed convection heat transfer in rotating vertical elliptic ducts

This paper presents an investigation into the solution of laminar mixed convective heat transfer in vertical elliptic ducts containing an upward flowing fluid rotating about a parallel axis. The coupled system of normalized conservation equations are solved using a power series expansion in ascending powers of rotational Rayleigh Number, Rat - a measure of the rate of heating and rotation as the perturbation parameter. The results show the influence of rotational Rayleigh number, Rat and modified Reynolds number, Rem on the temperature and axial velocity fields. The effect of Prandtl number, Pr, in the range 1 to 5, and eccentricity, e on the peripheral local Nusselt number are also reported. The mean Nusselt number is observed to be highest at duct eccentricity, e=0 for a given Prandtl number. However, results indicate insensitivity of peripheral local Nusselt number to Prandtl number at eccentricity, e=0.866, which is an important result to a designer of rotating vertical heat exchanger. The effect of eccentricity on the friction coefficient is also presented. The parameter space for the overall validity of the results presented is Rat RemPr<820.

Cold Simulation of Metalloids Transport in Blast Furnaces

Metalloids normally get transferred at the interface of metal droplets passing through the slag system in the dropping zone and at the slag‐metal interface in the hearth zone in the lower region of a blast furnace. In these high temperature processes, the mass transport being the rate‐controlling factor, the viscosity of the slag system determines the kinetics of the refining reactions accompanied by mass and heat transfer at the metal droplets and slag interface. Slag systems generally possess random network structures comprising internal regions of weak ordering. The presence of these regions may result in non‐Newtonian behaviour of the slag. The rheological characteristics of a fluid relating to its network structure is expressed in terms of the indices consistency (k') and flow behaviour (n').The extent of metalloids presence in hot metal is subjected to their residence time at the slag‐metal interface. The metal droplet descent through a surrounding fluid system has been studied and a co‐relation between drag Reynolds number and modified Reynolds number has been obtained. This correlation has been used to determine the drag velocity of a metal droplet falling through a slag system and the residence time distribution (RTD) of the metalloids at the slag‐metal interface in the lower region of the blast furnace.

Forced convection gaseous slip flow in circular porous micro-channels

Laminar forced convection of gaseous slip flow in a circular micro-channel filled with porous media under local thermal equilibrium condition is studied numer- ically using the finite difference technique. Hydrodynamically fully developed flow is considered and the Darcy-Brinkman-Forchheimer model is used to model the flow inside the porous domain. The present study reports the effect of several operating parameters (Knudsen number (Kn), Darcy number (Da), Forchhiemer number (�) , and modified Reynolds number (Re ∗ )) on the velocity slip and temperature jump at the wall. Results are given in terms of the velocity distribution, temperature distri- bution, skin friction (CfRe ∗ ), and the Nusselt number (Nu). It is found that the skin friction is increased by (1) decreasing Knudsen number, (2) increasing Darcy number, and (3) decreasing Forchheimer number. Heat transfer is found to (1) decrease as the Knudsen number, or Forchheimer number increase, (2) increase as the Peclet number or Darcy number increase. Nomenclature C Coefficient in the Forchheimer term Cf Skin friction coefficient Cp Constant pressure specific heat Cv Constant volume specific heat D Diameter of the circular channel Da Darcy number (K/er 2 ) h Local heat transfer coefficient K Intrinsic permeability of the porous medium Kn Knudsen number (λ/r0)

Oscillatory motion and wake instability of freely rising axisymmetric bodies

This paper reports on an experimental study of the motion of freely rising axisym- metric rigid bodies in a low-viscosity fluid. We consider flat cylinders with height h smaller than the diameter d and density ρb close to the density ρf of the fluid. We have investigated the role of the Reynolds number based on the mean rise velocity um in the range 80 ≤ Re = umd/ν ≤ 330 and that of the aspect ratio in the range 1.5 ≤ χ = d/h ≤ 20. Beyond a critical Reynolds number, Rec, which depends on the aspect ratio, both the body velocity and the orientation start to oscillate periodically. The body motion is observed to be essentially two-dimensional. Its description is particularly simple in the coordinate system rotating with the body and having its origin fixed in the laboratory; the axial velocity is then found to be constant whereas the rotation and the lateral velocity are described well by two harmonic functions of time having the same angular frequency, ω. In parallel, direct numerical simulations of the flow around fixed bodies were carried out. They allowed us to determine (i) the threshold, Recf1(χ), of the primary regular bifurcation that causes the breaking of the axial symmetry of the wake as well as (ii) the threshold, Recf2(χ), and frequency, ωf, of the secondary Hopf bifurcation leading to wake oscillations. As χ increases, i.e. the body becomes thinner, the critical Reynolds numbers, Recf1 and Recf2, decrease. Introducing a Reynolds number Re* based on the velocity in the recirculating wake makes it possible to obtain thresholds $\hbox{\it Re}^*_{cf1}$ and $\hbox{\it Re}^*_{cf2}$ that are independent of χ. Comparison with fixed bodies allowed us to clarify the role of the body shape. The oscillations of thick moving bodies (χ &lt; 6) are essentially triggered by the wake instability observed for a fixed body: Rec(χ) is equal to Recf1(χ) and ω is close to ωf. However, in the range 6 ≤ χ ≤ 10 the flow corrections induced by the translation and rotation of freely moving bodies are found to be able to delay the onset of wake oscillations, causing Rec to increase strongly with χ. An analysis of the evolution of the parameters characterizing the motion in the rotating frame reveals that the constant axial velocity scales with the gravitational velocity based on the body thickness, $\sqrt{((\rho_f-\rho_b)/\rho_f)\,gh}$, while the relevant length and velocity scales for the oscillations are the body diameter d and the gravitational velocity based on d, $\sqrt{((\rho_f-\rho_b)/\rho_f)\,gd}$, respectively. Using this scaling, the dimensionless amplitudes and frequency of the body's oscillations are found to depend only on the modified Reynolds number, Re*; they no longer depend on the body shape.

FLOWS WITH INERTIA IN A THREE-DIMENSIONAL RANDOM FIBER NETWORK

A fiber web is modeled as a three-dimensional random cylindrical fiber network. Nonlinear behavior of fluid flowing through the fiber network is numerically simulated by using the lattice Boltzmann (LB) method. A nonlinear relationship between the friction factor and the modified Reynolds number is clearly observed and analyzed by using the Fochheimer equation, which includes the quadratic term of velocity. We obtain a transition from linear to nonlinear region when the Reynolds numbers are sufficiently high, reflecting the inertial effect of the flows. The simulated permeability of such fiber network has relatively good agreement with the experimental results and finite element simulations.

Scaling in pipeline flow of Kaolin suspensions

Abstract The dimensionless analysis is applied to pipeline transportation of Kaolin suspensions. Experimental data were obtained for flow through pipes of different diameters. The rheological properties of materials under study were characterized by the Hershel–Bulkley equation. The method of constructing the modified Reynolds number, Rem for visco-plastic materials was discussed. It was shown that in a wide range of the Reynolds number (up to five decimal orders) the experimental data is well fitted by the standard f = 16/Rem dependence up to the laminar-to-turbulent transition point. The transition point (critical Reynolds number, Re m ∗ ) is a linear function of the pipe diameter. At Re m > Re m ∗ , the f(Rem) becomes dependent upon pipe diameter. Using the Re m ∗ value as a scaling factor, it is possible to construct a universal dependence of f Re m ∗ versus Re m / Re m ∗ , which is valid for the entire range of velocities and pipe diameters under study. In the laminar-to-turbulent transition zone this dependence approaches the asymptote f Re m ∗ = 14 and the pressure gradient becomes a quadratic function of average velocity.

A model for gas–solid flow in a horizontal duct with a smooth merge of rapid–intermediate–dense flows

Granular flow in the rapid flow regime is dominated by particle–particle collisions and the constitutive relations for the solid stress are obtained from the classic kinetic theory of granular flow. In the dense flow regime, on the other hand, particles interact via enduring contacts and the solid stress can be deduced from soil mechanics theories. In this paper, constitutive equations, recently proposed by Tardos et al. [2003. Slow and intermediate flow of frictional bulk powder in the Couette geometry. Powder Technology 131, 23–39.] has been incorporated in the simulation of gas–solid flow in a horizontal duct. These equations smoothly merge the rapid granular flow solution with the so-called “intermediate” regime (where both kinetic/collisional and frictional contributions might play a role) and reduce to Coulomb yield condition for slow frictional flow (shear rate → 0). The results of this new modelling approach have shown good qualitative agreement with the reported experimental observation on wide range of gas–solid flow conditions. In this study, we also present the definition of boundaries between rapid–intermediate–dense flow regimes based on the dimensionless shear rate ( λ ), and a modified Reynolds number ( Re). We have shown that the intermediate flow regime can be classified at approximately 0.1 < λ < 1.0 and 100 < Re < 3000 .

Hydrodynamic and Thermal Behavior of Gas Flow in Microchannels Filled with Porous Media

Laminar forced convection slip flow in parallel-plate microchannels filled with porous media is studied numerically. The governing momentum and energy equations are solved using the finite difference technique. The study is focused on the slip flow regime (i.e., for Knudsen numbers in the range 10−3 < Kn < 10−1 ) and the Darcy-Brinkman-Forchheimer model is used to model the flow inside the porous domain. The current study demonstrates the effects of Knudsen number Kn, Darcy number Da, Forchheimer number (Γ), and modified Reynolds number (Re ) on velocity slip and temperature jump at the wall(s). Results are given in terms of the product of the skin friction coefficient and the Reynolds number Cf Re* and the Nusselt number Nu. It is found that the skin friction is increased by: (i) increasing Darcy number and (ii) decreasing Knudsen number and/or Forchheimer number. Heat transfer is found to: (i) increase as the modified Reynolds number and/or Darcy number increases and (ii) decrease as the Knudsen number and/or Forchheimer number increases.

Power Consumption of Double-Bladed Kneaders of Various Shapes

The kneading process is widely used in various industries. When wet particles are mixed, the double-bladed kneader is one of the most useful instruments. We study the power consumption of a double-bladed kneader. The power is a very important factor in kneader designing and the examination of kneading conditions. In this paper, we propose an estimation method of the power consumption for double-bladed kneaders of various geometrical con-figurations based on the relation of the power number, NP, and the modified Reynolds number, Re′. The influences of the vessel type, blade shape, wet particle volume and blade thickness on the power index, β, and the equivalent yield stress, (τc/B) were investigated. The influence of the geometrical configuration of a kneader on β and (τc/B) was explained using the dimensionless wet particle volume, V/v, and the dimensionless effective shearing area, S/s, respectively. These relations were correlated as a power expression. The equations obtained could be used to estimate the values of β and (τc/B) with errors of ±5% and ±10%, respectively. The applicability of these equations was examined in experiments using wet CaCO3 particles and wet kaolin particles. Furthermore, in this study, the power consumption of various kneaders was estimated concretely. As a result, it was shown that the power consumption could be estimated with good accuracy.