A numerical study of the accelerating motion of a dense rigid sphere in non-newtonian power law fluids

The equations of motion of an accelerating sphere falling through non-Newtonian fluids with power law index n in the range 0.2 ≤ n ≤ 1.8 were integrated numerically using the assumption that the drag on the sphere was a function of both power law index and terminal Reynolds number, Re_t For 10^?2 ≤ Re_t ≤ 10³ both dimensionless time and distance travelled by the sphere under transient conditions showed a much stronger dependence on the flow behaviour index, n, for shear-thinning than for shear-thickening fluids. The form of this dependence is investigated here. Furthermore, results in four typical shear-thinning fluids suggested a strong correlation between the distance and time travelled by the sphere under transient conditions and the value of the fluid consistency index. The analysis reported herein is, however, restricted to dense spheres falling in less dense fluids, when additional effects arising from the Basset forces can be neelected.     

Steady Flow of Power Law Fluids across a Circular Cylinder

The momentum equations describing the steady cross‐flow of power law fluids past an unconfined circular cylinder have been solved numerically using a semi‐implicit finite volume method. The numerical results highlighting the roles of Reynolds number and power law index on the global and detailed flow characteristics have been presented over wide ranges of conditions as 5 ≤ Re ≤ 40 and 0.6 ≤ n ≤ 2. The shear‐thinning behaviour (n < 1) of the fluid decreases the size of recirculation zone and also delays the separation; on the other hand, the shear‐thickening fluids (n > 1) show the opposite behaviour. Furthermore, while the wake size shows non‐monotonous variation with the power law index, but it does not seem to influence the values of drag coefficient. The stagnation pressure coefficient and drag coefficient also show a complex dependence on the power law index and Reynolds number. In addition, the pressure coefficient, vorticity and viscosity distributions on the surface of the cylinder have also been presented to gain further physical insights into the detailed flow kinematics.     

Sedimentation of a sphere along the axis of a long square duet filled with non-newtonian liquids

Based on extensive experimental results, it is shown that the retardation effect caused by the confining walls on the free settling velocity of a sphere is smaller with square walls than that with cylindrical boundaries. This is true for both Newtonian and power law fluids, provided the particle Reynolds number is small (< about 5). The values of the wall factor for Newtonian liquids are in excellent agreement with theory (up to R / L ≤ 0.1) while those for power law fluids have been correlated empirically via a linear relationship. The results reported here encompass the following ranges of conditions: 1 ≥ n ≥ 0.7; Re < 15 and 0.024 < R/L < 0.238.     

Particle trajectories and terminal velocities in vertically oscillating fluids

The trajectories and terminal velocities of particles in vertically oscillating fluids have been studied by obtaining analytic and numerical solutions to the nonlinear Langevin equation representing a superposition of forces arising from particle acceleration, displaced fluid acceleration, buoyancy and particle-fluid drag as represented by an n^th-power drag law. In vibrating fluids the directional particle velocities are found to be lower than in stationary fluids for drag exponents n > 1 and correspondingly higher for 0 < n < 1. The engineering significance of the results is discussed in relation to hold-up, separation and transport phenomena in vertically oscillating multiparticle-fluid systems.     

Power‐Law Fluid Flow Over a Sphere: Average Shear Rate and Drag Coefficient

Based on the consideration of the rate of mechanical energy dissipation, an expression for the average shear rate for a sphere falling in a power‐law fluid in the creeping flow regime has been deduced. The average shear rate in a power‐law fluid (n<1) appears to be higher than that in an equivalent Newtonian fluid. This in turn has been combined with the numerical predictions of drag coefficient (up to Reynolds number of 100) of a sphere to develop a generalized drag correlation for power‐law liquids encompassing both n > 1 and n < 1 which appears to apply up to much higher values of the Reynolds number. The available experimental data have been used to demonstrate the reliability and accuracy of the new correlation for shearthinning liquids. Also, in the limit of n = 1, this expression reproduces the standard drag curve with a very high accuracy.     

Axial dispersion in laminar flow of polymer solutions through coiled tubes

The results of an experimental study on axial dispersion in laminar flow of non-Newtonian fluids through helical coils are reported. The ranges of variables covered are 10.5 ≤ λ ≤ 220, 0.6 ≤ n ≤ 1.0, 0.1 < N_Regn < 140, and 0.04 < τ > 2.2. The condition for the applicability of Taylor's dispersion model is also reported. It is found that coiling results in a dispersion reduced over that in a straight tube.     

Steady Two‐Dimensional Non‐Newtonian Flow Past an Array of Long Circular Cylinders up to Reynolds number 500: A Numerical Study

The free surface and zero vorticity cell models have been combined with the equations of motion to investigate numerically the steady flow of incompressible power‐law (shear‐thinning and shear‐thickening) fluids across banks of long cylinders. The equations of motion in the stream function/vorticity formulation have been solved numerically using a second order accurate finite difference method to obtain extensive information on the behaviour of the drag coefficient, surface vorticity distribution, streamlines and iso‐vorticity patterns, for high Reynolds numbers (Re = 50 500) and using a wide range of power‐law index (0.3 ≤ n ≤ 2.0), and porosity (0.4 ≤ e ≤0.9) values. The behaviour of the aforementioned parameters at low Reynolds numbers has also been investigated and validated using theoretical and numerical work from the literature. The results reported here enable extension of the limits of creeping flow behaviour up to Re = 50 for fluids with highly shear‐thickening characteristics under low porosity conditions.     

Theoretical Study of Equilibrium Bipolar Charge Distribution on Nonuniform Primary Straight Chain Aggregate Aerosols

A general theory describing the equilibrium bipolar charge distribution for straight chain aggregate aerosols consisting of primary spheres of different diameters was derived from a theory previously developed for linear chain aggregate of uniform spheres. The present theory is based on the assumptions that (1) the individual primary particles of a straight chain aggregate are charged independently, (2) the probability that a particular primary particle has acquired q elementary charges is governed by the Gaussian distribution predicted by Boltzmann's law, based on particle size; and (3) the resultant charge of a straight chain aggregate is the algebraic sum of the charges carried by the constituent primary spheres. The present theory can be stated as follows: The equilibrium bipolar charge distribution of straight chain aggregate aerosols with nonuniform primary spheres can be expressed by Boltzmann's law with an equivalent diameter such that d_eL = Σ ⁿ _i=1 d_i . The limitations imposed by the assumptions are also discussed.     

Terminal velocity of porous spheres

Terminal velocity of porous spheres was experimentally measured for a Reynolds number range of 0.2 to 120 for a normalized sphere radius, β = R/R of 15.6 to 33, where R and k are the sphere radius and permeability, respectively. The drag coefficient for 15 < β < 33 was found to be C_D = 24Ω/Re [1 + 0.1315 Re^{(0.82 - 0.05w)}] for 0.1 < Re ≤ 7 and C_D = 24Ω/Re [1 + 0.0853 Re^{(1.093 - 0.105w)}] for 7 < Re < 120 with w = log₁₀Re where Re is the sphere Reynolds number and Ω=2β² [1 - (tanh β/β)] / 2β² + 3[1 - tanh β/β)] At high Reynolds numbers, it was found that the porous sphere terminal velocity was less affected by the container walls than for the case of an impermeable sphere. However, at very low Reynolds numbers, the wall effects were found to be similar for both the permeable and the impermeable spheres.     

Characterisation of inelastic power-law fluids using falling sphere data

Two simple methods are presented for the characterization of inelastic power law fluids from falling sphere data. The methods involve the application of shear rate or shear stress correction factors which have been derived theoretically using Slattery's solution for creeping flow about spheres. Flow curves obtained using these methods are in excellent agreement with those measured on a Weissenberg rheogoniometer for 0.83 ≤ n ≤ 1.0. The experimentally determined drag coefficients are found to be in good agreement with the predictions of Slattery's creeping flow first approximation solution. The wall correction factors of Faxen and Francis appear to be valid for inelastic non-Newtonian fluids up to a diameter ratio of at least 0.08.     

Unified entry length for newtonian and power–law fluids in laminar pipe flow

A numerical method based on finite differencing is used for investigating the steady–state entrance region laminar flow of incompressible Newtonian and power–law fluids in a circular pipe. The Solution method is validated by comparing the results for Newtonian fluids with those reported in the literature. For power–law fluids, the entry length results are compared with other approximate solutions in the literature. On the basis of the calculated results, a generalized entry length ξ99 = 0.056 is shown to be valid for the laminar flow at Re > 200 of both Newtonian and power–law fluids with 0.75 < n < 1.5.     

Hydrodynamic drag forces on two porous spheres moving along their centerline

This paper numerically evaluates the hydrodynamic drag force exerted on two highly porous spheres moving steadily along their centerline (sphere #1 and sphere #2) through a quiescent Newtonian fluid over a Reynolds number ranging from 0.1 to 40. At creeping flow limit, the drag forces exerted on both spheres were identical. At higher Reynolds numbers the drag force on sphere #1 was higher than sphere #2, revealing the shading effects produced by sphere #1 on sphere #2. At dimensionless diameter (β, =d_f/2k^0.5, d_f and k are floc diameter and interior permeability, respectively) >20, the spheres can be regarded nonporous. At β<20, the drag forces dropped. At β<2, the drag forces approached “no-spheres” limit. An increased size ratio of two spheres (d_f1/d_f2) would increase the drag force on sphere #1 and reduce that on sphere #2. At increasing β for both spheres, the drag force on sphere #2 was increased because of the more difficult advective flow through its interior, and at the same time the drag was reduced owing to the stronger wake flow produced by the denser sphere #1. The competition between these two effects leads to complicated dependence of drag force on sphere #2 on β value. These effects were minimal when β became low. Two identical spheres could move steadily along their centerline. At higher Reynolds number, the two spheres would move closer because of the incorporation of inertia force. For spheres of different diameters, the sphere # 2 would move faster than sphere #1 regardless of their size ratio and β value. This occurrence yielded efficient coagulation when two porous spheres were moving in-line.     

Laminar dispersion of polymer solutions in helical coils

In the present study the step response experiments were carried out with power law fluids in two helical coils to examine the suitability of axial dispersed plug flow model in describing the laminar dispersion of non-Newtonian fluids in helical coils. The ranges of variables covered are 10 ≤ λ ≤ 100,0.01 ≤ N_Regen ≤ 2.5,0.001 ≤ N_De ≤ 0.77 and 0.035 ≤ τ ≤ 1.33. It is found that coiling results in reduced dispersion to that in a straight tube.     

An experimental study of non-newtonian fluid flow through fixed and fluidized beds of non-spherical particles

Extensive measurements of pressure drop in fixed beds, minimum fluidization velocity and expansion characteristics for beds of non-spherical particles are reported in the following ranges of conditions: 10^-3 ≤ Re ≤ 20; 0.66 ≤ n ≤ 1 and 0.41 ≤ ? ≤ 0.75. Based on an analysis of these results, it is illustrated that the existing frameworks originally developed for Newtonian fluid flow through beds of spherical particles are also satisfactory for power law fluid flow through beds of non-spherical particles, provided a volume equivalent diameter modified by a sphericity factor and a modified Reynolds number are used instead of their usual definitions.     

The Effects of Power Law Fluid Rheology on Coil Heat Transfer for Agitated Vessel in the Transitional Flow Regime

A CFD model of heat transfer from power‐law fluids to helical cooling coils in the transitional flow regime of a baffled tank mixed with a pitched blade turbine was developed with Fluent^TM. The model captured local temperature and velocity gradients. Simulations were run, varying Re, Pr, K and n. The results indicate that a Sieder‐Tate type correlation, with the exponent on and the coefficient in front of the Reynolds number being a function of n, is recommended for estimating ho. Also, a new two coil bank design was found to be more efficient when 450 < Re < 650.     

Effects of Viscous Dissipation on Heat Transfer between an Array of Long Circular Cylinders and Power Law Fluids

The free surface model has been combined with the equations of motion and of thermal energy to investigate the role of viscous dissipation on heat transfer between banks of long cylinders and power law (shear‐thinning and shear‐thickening) fluids. The equations of motion cast in the stream function/vorticity formulation have been solved numerically using a second‐order accurate finite difference method to obtain extensive information on the behaviour of local and surface‐averaged Nusselt numbers over a range of Reynolds numbers 1 – 500, for a wide range of power law indices (0.4 ≤ n ≤ 2.0), Brinkman numbers (0 ≤ Br ≤ 5) and Prandtl numbers (Pr = 1, 1000) at two representative solid volume fractions corresponding to the porosities of e = 0.4 and 0.9. Two different thermal boundary conditions are considered at the cylinder surface: constant temperature (CT) and constant heat flux (CHF). The results presented herein provide a fundamental knowledge about the influence of viscous dissipation on the heat transfer characteristics. The results reported herein further show that the effect of Brinkman number on heat transfer is strongly conditioned by the thermal boundary condition, Prandtl number and the power law index.     

Characterisation of inelastic poweriaw fluids using falling sphere data

Two simple methods are presented for the characterization of inelastic power law fluids from falling sphere data. The methods involve the application of shear rate or shear stress correction factors which have been derived theoretically using Slattery's solution for creeping flow about spheres. Flow curves obtained using these methods are in excellent agreement with those measured on a Weissenberg rheogoniometer for 0.83 > n > 1.0. The experimentally determined drag coefficients are found to be in good agreement with the predictions of Slattery's creeping flow first approximation solution. The wall correction factors of Faxen and Francis appear to be valid for inelastic non-Newtonian fluids up to a diameter ratio of at least 0.08.     

Hindered settling in non-newtonian power law liquids

New experimental results on the hindered settling of model glass bead suspensions in non-Newtonian suspending media are reported. The data presented encompass the following ranges of variables: 7.38 × 10^?4 ≤ Re_1∞ ≤ 2; 0.0083 ≤ d/D ≤ 0.0703; 0.13 ≤ C ≤ 0.43 and 1 ≥ n ≥ 0.8. In these ranges of conditions, the dependence of the hindered settling velocity on concentration is adequately represented by the corresponding Newtonian expressions available in the literature. The influence of the power law flow behaviour index is completely embodied in the modified definition of the Reynolds number used for power law liquids.     

Drag coefficients of irregularly shaped particles

The steady-state free-fall conditions of isolated groups of ordered packed spheres moving through Newtonian fluids have been studied experimentally. Measurements of the drag coefficients are reported in this paper for six different geometrical shapes, including isometric, axisymmetric, orthotropic, plane and elongated conglomerates of spheres. From these measurements, a new and accurate empirical correlation for the drag coefficient, C_D, of variously shaped particles has been developed. This correlation has been formulated in terms of the Reynolds number based on the particle nominal diameter, Re, the ratio of the surface-equivalent-sphere to the nominal diameters, d_A/d_n, and the particle circularity, c. The predictions have been tested against both the experimental data for C_D collected in this study and the ones reported in previous works for cubes, rectangular parallelepipeds, tetrahedrons, cylinders and other shapes. A good agreement has been observed for the variously shaped agglomerates of spheres as well as for the regularly shape particles, over the ranges 0.15<Re<1500, 0.80<d_A/d_n<1.50 and 0.4<c<1.0.     

Dependence of particle fluctuation velocity on gas flow, and particle diameter in gas fluidized beds for monodispersed spheres in the Geldart B and A fluidization regimes

In this paper we present new experimental data on the steady-state, mean squared, fluctuation velocity, or granular temperature, of Geldart B polymer, glass, nickel, and stainless steel monodispersed spheres averaged over the wall of a gas fluidized bed, as a function of gas flow and sphere diameter. The granular temperature is obtained by Acoustic Shot Noise technology—namely power spectral analysis of the steady state vibrational energy of the wall excited by random sphere impact, and calibrated by hammer excitation over the wall. The new data extends to polymer and metallic spheres the experimental discovery of a 1996 paper of Cody et al. that the fluctuation velocity of Geldart B glass spheres when scaled to the gas superficial velocity, U_s, is inversely proportional to sphere diameter, directly proportional to a fundamental length scale, D_o^B, and is a universal function of U = (U_s / U_mf). We also demonstrate that the new data is consistent with the diameter dependence of the fluctuation velocity that can be derived from both the 1997 paper of Menon and Durian, who measured random sphere motion near the wall through the spectroscopy of scattered laser light, and the 1992 paper of Rahman and Campbell, who measured the average granular pressure of random sphere impact on a porous steel membrane. While the inverse scaling of the fluctuation velocity with sphere diameter, and the existence of a fundamental length scale for gas fluidization, D_o^B, had not been a feature of any published fundamental model, or computer simulation, of the steady state granular temperature of spheres in gas fluidized beds, we show that it is a feature of two recent dense kinetic fluidization models published in 1999, by Buyevich and Kapbasov, and Koch and Sangani. Both theories implicitly define a fundamental length scale for the fluctuation velocity, D^? = (μ_f² / ρ_p²g)^1 / 3, where ρ_p is the sphere density, μ_f is the gas viscosity, and g is the laboratory gravitational field. The new data for polymer, glass, nickel and stainless steel spheres presented in this paper, defines D_o^B = (56 ± 2)D^?. We use the Anderson-Jackson stability model to show that the length scale D_o^B, also defines a stability length scale, such that for D < D_o^B(D > D_o^B), the uniform dense phase of the fluidized bed is stable (unstable), against one dimensional, first order fluctuations in sphere concentration. The length scale, D_o^B is thus the theoretical equivalent to the empirical scaling length introduced by Geldart, D_B/A, to distinguish spheres (D > D_B/A) that bubble at fluidization, from spheres (D < D_B/A) that fluidize before bubbling. Finally, we present new experimental data, on the remarkable changes in the granular temperature, bed expansion, and bed collapse time, between Geldart B and Geldart A monodispersed glass spheres, and compare that data to granular temperature, and bed expansion, for Geldart A rough, non-spherical, log-normal dispersed diameter catalytic particles.