The effect of quantum vibrations on electrochemical outer sphere redox reactions

The influence of an inner sphere vibration with a frequency in the region 10¹³ s^?1 ≤ ω ≤ 4 × 10¹⁴ s^?1 on simple electrochemical redox reactions is examined. The numerical calculations, based on first order perturbation theory and the Frank—Condon approximation, exhibit the following effects: the energy of activation decreases with temperature due to an increase in tunnel transitions of the system; the transfer coefficient is temperature dependent; the curvature of Tafel lines is greater at lower temperatures. All of these effects are of a quantum mechanical nature, and disappear both for very high and very low frequencies. These results are compared with classical and semi-classical approximations. At high overpotentials, the calculations predict limiting currents corresponding to activationless transfer. The magnitude of the limiting current is determined by the electronic matrix element and is independent of the model Hamiltonian for the solvation sphere.     

Free convection in power-law fluids from a heated sphere

In this work, the governing field equations describing heat transfer from a heated sphere immersed in quiescent power-law fluids have been solved numerically. In particular, consideration has been given to elucidate the role of Grashof number (Gr), Prandtl number (Pr) and power-law index (n), on the value of the Nusselt number (Nu) for a sphere in the natural convection regime. Further insights are provided by presenting streamline and constant temperature contours. The results presented herein encompass the following ranges of conditions: 10≤Gr≤10⁷; 0.72≤Pr≤100 and 0.4≤n≤1.8 thereby covering both shear-thinning and shear-thickening types of fluid behaviours. Broadly, all else being equal, shear-thinning behaviour can enhance the rate of heat transfer by up to three-fold where as shear-thickening can impede it up to ～30?40% with reference to that in Newtonian fluids. The paper is concluded by presenting detailed comparisons with the scant experimental data and the other approximate treatments of this problem available in the literature.     

Mass transfer around freely moving active particles in the dense phase of a gas fluidized bed of inert particles

The mass transfer coefficient around freely moving active particles under bubbling/slugging fluidized bed conditions was measured in a lab-scale reactor. The technique used for the measurements consisted in the oxidation reaction of carbon monoxide at over one or few Pt catalyst spheres immersed in an inert bed of sand. It was shown that this technique is simple and accurate, and allows to overcome most of the difficulties and uncertainties associated with other available techniques. The experimental campaign was carried out by varying the fluidization velocity (0.15-0.90 m/s), the active particle size (1.0-10.0 mm) and the inert particle size (0.1-1.4 mm). Results were analyzed in terms of the particle Sherwood number. Experimental data showed that Sh is not influenced by the fluidization velocity and by a change of regime from bubbling to slugging, whereas it increases with a square root dependence with the minimum fluidization velocity and with the active particle size. These results strongly suggest that the active particles only reside in the dense phase and never enter the bubble/slug phase. Data were excellently fitted by a Frössling-type correlation:

Sh=2.0·ε_mf+K·(Re_mf/ε_mf)^1/2·Sc^1/3     

Effectiveness factors of nth order kinetics in trickle-bed reactors

In trickle-bed reactors conventional solutions of effectiveness factors are not applicable because parts of the external catalyst surface may be covered by liquid and others by gas. In this paper approximate solutions of effectiveness factors are systematically compared with numerical solutions for a sphere model with symmetrical boundary conditions and nth order kinetics. The finite difference method with successive-over-relaxation is adopted for the computations. Comparison of the results indicates that simple approximate overall effectiveness factors give good results for both linear and nonlinear kinetics.     

Direct numerical simulations of a freely falling sphere using fictitious domain method: Breaking of axisymmetric wake

In the present paper, numerical simulations of the wake generated by a freely falling sphere, under the action of gravity, are performed. Simulations have been carried out in the range of Reynolds numbers from 1 to 210 for understanding the formation, growth and breakup of the axisymmetric wake. The in-house code used is based on a non-Lagrange multiplier fictitious-domain method, which has been developed and validated by Veeramani et al. (2007). The onset of instability in the wake and its growth along with the dynamic behavior of a settling sphere is examined at Reynolds number (Re) of 210. It is found that at the onset of instability the sphere starts to rotate and gives rise to a lift force due to the break of the axisymmetry in the wake which in turns triggers a lateral migration of the sphere. The lift coefficient of a freely falling sphere is 1.8 times that of a fixed sphere at a given sphere density of 4000 kg m^?3 and sphere to fluid density ratio of 4. This is attributed to the Robin's force which arises due to the rotation of the sphere. At this Reynolds number (Re=210) a double threaded wake is observed, which resembles the experimental observations of Magarvey and MacLatchy (1965).     

Hydrodynamic force between two hard spheres tangentially translating in a power-law fluid

The hydrodynamic interaction between two hard spheres tangentially translating in a power-law fluid is investigated. By considering the gap between the two spheres being sufficiently small such that the Reynolds’ lubrication theory applies, an analytical equation to the pressure in the gap is obtained using truncated Fourier series. To a good approximation, the pressure equation can be further simplified. The simplified approximate equation over-predicts the pressure for shear thickening fluid (n>1) but under-predicts the pressure for shear-thinning fluid (n<1). However, the errors in the predicted tangential force and moment are relatively small. In particular, for a Newtonian fluid, the accurate solution and the simplified approximate solution degenerate to the asymptotic solution of Goldman et al. [1967. Slow viscous motion of a sphere parallel to a plane wall-motion through a quiescent fluid. Chemical Engineering Science 22, 637-651.] and O’Neill and Stewartson [1967. On the slow motion of a sphere parallel to a nearby plane wall. Journal of Fluid Mechanics 27, 705-724.]. Both solutions predict that for shear thickening fluid (n>1), the hydrodynamic force converged in the inner region of the gap between the two spheres and the contribution from the outer region is sufficiently small. For shear thinning fluid (n<1), the contribution from the outer region is also significant.     

Intermolecular interactions and formation of the hydration sphere in phosphonic acid model systems as an approach to the description of vinyl phosphonic acid based polymers

Molecular model systems based on propyl phosphonic acid (ppa) were studied by means of density functional theory calculations in order to describe the acid-acid interaction and the formation of the hydration sphere. The formation of ppa dimers is reported and the energetic difference between two dimer structures is presented. The hydration sphere of ppa was represented by model systems ppa(H₂O)_n, for which the system with n=4 formed the first hydration sphere (h¹), while n=7 can be considered a good approximation to the complete inner hydration sphere around the phosphonic acid group. The study of the ppa-H⁺ (H₂O)_n model systems showed an interesting structural behavior comparatively to the ppa(H₂O)_n systems. The protonated acids exhibited equivalent phosphorous-oxygen bonds and a general molecular structure is proposed to represent these protonated species.     

Simultaneous absorption of two gases accompanied by a complex chemical reaction: approximate solutions

Approximate solutions to the problem of simultaneous absorption of two gases in a liquid accompanied by a complex chemical reaction have been presented based on the film theory. Two approximate profiles, a nonlinear exponential profile and a trigonometric profile, for the concentration of each of the gaseous species in the film have been used in analysing the problem. The complex scheme considered is:

For the exponential case two approximations have been considered: [1] in which the higher order terms are included, and [2] in which the higher order terms are neglected.The results obtained using the two profiles have been compared with numerical solutions for the film theory in the range of

from 1 to 3. The results show that both the approximations yield solutions close to the numerical, in particular case [1] of the exponential approximation. Some special cases have then been considered followed by a discussion of an industrially important system: simultaneous absorption of ethylene and chlorine in water to give ethylene chlorohydrin.     

Mass transfer to large particles in fluidised beds of smaller particles

A theory is proposed for predicting the transfer of a gas through a fluidised bed of small particles to a large particle. It is proposed that non steady-state mass transfer of the gas occurs by two mechanisms: (i) mass transfer of gas in clusters or packets of the smaller particles approaching the large particle; and (ii) gas convection. The theory developed enables prediction of the Sherwood number (N_sh, the dimensionless mass transfer coefficient) for a large particle, diameter d: N_sh=2ε_mf+$\text{4}\text{g3}\text{mfd}\text{(U}\text{mf}\text{ε}\text{mf}\text{+}\text{uinb}$)/π D_A${}^1{}^2$ where U_mf is the minimum fluidising velocity, εmf is the bed voidage at U_mf-0u_b is the mean bubble rise velocity and D_A is the gas diffusivity. This equation is shown to be in excellent agreement with Sherwood numbers determined from combustion experiments in which single large particles of petroleum coke were burned in air fluidised beds over a wide range of operating conditions. It is also shown that predictions using this expression are in close agreement with those from an empirical expression previously proposed by the autho     

Determination of the collision frequency between bubbles and particles in flotation

Collision efficiency for a spherical bubble rising in a uniform concentration of small non-inertial particles is studied by direct numerical simulations (DNS). The Stokes number of the particles is negligibly small so that the particle trajectories follow the streamlines. The effect of the bubble interface contamination is studied for the flow surrounding the bubble using the spherical cap model. Numerical results are obtained for a wide range of bubble Reynolds number (based on bubble diameter d_b) ranging from 0.01 to 1000 and for different angles of contamination ranging from 0^° to 180^°. The collision efficiency is found to be increased with the Reynolds number and significantly decreased with the level of contamination. Correlations of the numerical results are proposed for efficiencies versus d_p/d_b (d_p being the particle diameter), bubble Reynolds number and interface contamination degree. For clean (respectively, fully contaminated) spherical bubbles, the efficiency evolves as d_p/d_b (respectively (d_p/d_b)²) whatever the bubble Reynolds number and the particle size. For partially contaminated bubbles, efficiency can be scaled with d_p/d_b or (d_p/d_b)² depending on both the level of contamination and the particle size.     

Low to moderate Peclet mass transport in assemblages of spherical particles for a realistic adsorption-reaction-desorption mechanism

A theoretical model and the associated numerical simulations for the mass transport from a moving Newtonian fluid to an assemblage of spherical solid absorbers are presented here. In particular, we present results from the numerical solution of the convection-diffusion equation in the simplified sphere-in-cell geometry and in stochastically constructed 3-D spherical particle assemblages for low to moderate Peclet numbers (Pe < 100) and relatively high porosities (? > 0.7). A realistic adsorption/reaction/desorption mechanism is used to describe the adsorption of diluted mass on the particles surface as opposed to the assumption of instantaneous and Langmuir-type adsorption that has been adopted in previous works. We also attempt to compare the effect of considering different sorption mechanisms in terms of adsorption efficiency. In all cases, the adequacy of the simplified sphere-in-cell approach is tested against the predictions from the numerical study in sphere assemblages. It is found that higher adsorption efficiencies correspond to lower porosities while increasing Peclet numbers lead to lower λ₀ values. Finally, it is shown that the assumption of instantaneous adsorption leads to severe overestimation of the adsorption efficiency in comparison with that obtained by using the more realistic adsorption-reaction-desorption model.     

A heat and mass transfer study—analysis of continuous processes for the coloration of polyester fabric. I. Heat transmission to the fabric system—fabric,fiber, and dye particles

The three continuous processes (thermosol, high temperature steaming, and heat transfer printing) for the coloration of polyester and the involved equipment are briefly reviewed. A simple model for the transient heating of a body is developed, the model [eq.(22)] comprises a specific area of transfer parameter (A_s): area through which transport (of heat) takes place per unit mass of the body. This model is used in order to estimate the relative heating rates for the different geometrical approximations present in the fabric system: plate (fabric), cylinder (fiber), and sphere (dye particle). Results obtained from the more exact numerical analysis solution are used for investigating the limitations of the simple heat transfer model.     

Evaporation from a nonspherical aerosol particle situated in an absorbing gas

The problem of an aerosol particle evaporating in an infinite expanse of an absorbing gas is considered. The relevant Helmholtz equation (resulting from the steady-state diffusion equation with an absorption term included) with density jump boundary conditions is converted into a boundary integral equation via the use of the Green’s function. The resulting integral equation is valid for particles of arbitrary shape. Explicit numerical results for the local and average evaporation rates are reported for several axisymmetric particles for a range of values of the dimensionless absorption parameter (λ²), where λ is the ratio of the radius of the particle (a) to the diffusion length (l). Here, the diffusion length is defined as l=[D/(vΣ_a)]^1/2, in which v (cm s^-1) is the average thermal speed of the vapor molecules, Σ_a (cm^-1) is the cross-section for absorption of the vapor by the gas, and D (cm² s^-1) is the diffusion coefficient of the vapor in the gas. Our numerical results for the local and average evaporation rates for a sphere exhibit excellent agreement with the corresponding analytical values (maximum deviation <0.40%). We find that the evaporation rate increases with increasing absorption and that this increase depends on the degree of departure of the particle from a spherical shape. The jump distance has a large impact in that it significantly lowers the evaporation rates as it increases in magnitude. It should be remarked that the results of this paper are also directly applicable to the problem of either neutrons or photons undergoing diffusion from a source situated in an absorbing medium.     

Robust Linear Interpolation and Extrapolation of Stationary Time Series in Lp

To deal with uncertainty of the spectral distribution, we consider minimax interpolation and extrapolation problems in L^p for stationary processes. The interpolation and extrapolation problems can be regarded as a linear approximation problem on the unit disk in the complex plane. Although the robust one-step-ahead predictor and interpolator has already been considered separately in the previous literature, we give two conditions for the uncertainty class to find the minimax interpolator and extrapolator in the general framework from both the point of view of the observation set and the point of view of evaluation on the interpolation and extrapolation error under the L^p-norm. We show that there exists a minimax interpolator and extrapolator for the class of spectral densities ε-contaminated by unknown spectral densities under our conditions. When the uncertainty class contains spectral distribution functions which are not absolutely continuous to the Lebesgue measure, we show that there exists an approximate interpolator and extrapolator in L^p such that its maximal interpolation and extrapolation error is arbitrarily close to the minimax error when the spectral distributions have densities. Our results are applicable to the stationary harmonizable stable processes.     

A NON-DRIVING-FORCE APPROXIMATION FOR INTRAPARTICLE DIFFUSION IN CYCLIC ADSORPTION AND DESORPTION WITH SHORT CYCLE TIMES

Simple approximations for intraparticle diffusion rate are obtained for cyclic adsorption-desorption subject to step changes at a cyclic steady state. For slow cycles (θ_c > 0.1), a linear driving-force approximation with a coefficient π² gives good prediction whereas for fast cycles (θ_c ≤ 0.l) a non-driving-force approximation with a constant coefficient (independent of cycle time θ_c) gives satisfactory results. The constant coefficient is an advantage of the non-driving-force approximation over the linear driving-force approximation in which the coefficient increases sharply with decreasing cycle time.     

Non-Newtonian fluidisation of spherical particles

This work presents the development of a new model based on the submerged object concept in order to characterise the expansion of spherical particles fluidised by non-Newtonian purely viscous liquids, especially at intermediate and high bed void fractions values. In order to take the interactions between the particles into account a parameter called hydraulic tortuosity is introduced which is a function of the Reynolds number and porosity. A method allowing its evaluation is proposed. Based on the test of a lot of experimental data from our laboratory and from previous works, this model is shown to give satisfactory results for porosities values larger than 0.6. In order to cover the entire range of porosities corresponding to the particles expansion a capillary-type model is proposed for the lower values of the porosities. This model is found to give satisfactory results in the range of bed void fraction comprised between the value corresponding to the minimum of fluidisation ε_mf and ε=0.65. The combination of the two models tested on 21 independent sets of data concerning viscoinelastic shear-thinning liquids leads to a mean error of 16%. It is also shown to give accurate predictions for Newtonian liquids. A comparison is also presented with the prediction of some of the widely used equations available in literature.     

On non-Newtonian flow past a sphere

The approximate solutions for flow of and Ostwald-de Waele fluid past a sphere at R_e·0 = 60 and 1 ? n ? 0·8 are obtained by the use of an extended method of moments. As n decreases, (1) friction drag decreases, (2) pressure drag increases for flow past a blunt body, (3) total drag increases for flow past a sphere, (4) wake length increases for flow past a sphere, (5) separation point moves forward for flow past a sphere.     

On the quantification of energy dissipation in the impeller stream of a stirred vessel from fluctuating velocity gradient measurements

The turbulence energy dissipation rate (ε) in the impeller stream of a vessel of diameter stirred by a Rushton turbine of diameter D=T/3 was directly measured with particle image velocimetry (PIV). Both 2-D and 3-D PIV techniques were employed to measure the mean velocities, Reynolds stresses and ε in the vessel for Reynolds numbers of 15 000-40 000. ε was determined directly from measurements of the fluctuating velocity gradients by analysing the PIV images with a resolution of . The values of the normalised ensemble-averaged dissipation rate (ε/N³D²) in the impeller stream were in the range 5-10. The measured fluctuating velocity gradients compared well with similar data obtained using a four-channel laser anemometer. The results are also compared with those of earlier works employing non-direct methods to estimate ε and show that some of these methods yield comparable values, although the spread of the some of the data previously reported is significant. The present results show the feasibility of direct measurement of the ε distribution with PIV and provide useful information for the design of mixing processes as well as for its more accurate estimation in future work.     

Angle-resolved stereo-PIV measurements close to a down-pumping pitched-blade turbine

The present work employs a stereoscopic-PIV technique to obtain angle-resolved fields of all three velocity components close to a T/3, 45^° down-pumping pitched-blade turbine operated at 300 rpm in a 0.29 m diameter vessel. The measurements were made at blade angles 7.5^° apart, with 300 measurements taken at each blade position, in order to calculate angle-resolved mean velocity fields and turbulence quantities. Turbulent kinetic energy (k) distributions were obtained using (i) a pseudo-isotropic approximation, from two velocity components and (ii) a full calculation from all three velocity components. The two calculation methods for k yielded similar results, indicating that data from 2-D PIV measurements yield reasonable estimates of the turbulence kinetic energy. The tangential velocity components at the impeller discharge from PIV were in good agreement with data from LDA analysis. A kinetic energy balance across the impeller was performed (i) rigorously and (ii) using approximations which neglected second- and higher-order velocity cross-correlations. Both analyses show that around 44% of the total power consumed by the impeller is dissipated in the impeller region. The average rate of dissipation of kinetic energy is about 40 times higher in the impeller region than the volume-average dissipation rate in the whole vessel.