A parameter characterizing the geometry of expanded metal. II. Natural convection mass transfer at expanded metal electrodes

Free convective mass transfer rates at vertical electrodes of expanded metal were measured by the electrochemical method. Electrode height and electrolyte concentration were varied and the dependence of the expanded metal on the geometry and on the mesh orientation with respect to the vertical direction was investigated. A single equation was developed to correlate all the results. Besides the generalized dimensionless groups for natural convection the correlation includes a parameter characterizing the geometry of the expanded metal. The correlation also represents free convective mass transfer results obtained by other investigators with vertical mesh electrodes.Nomenclature a width of narrow space - A mean mesh aperture - c ₀ bulk concentration - d cavity diameter - d _p particle diameter - D diffusivity - g acceleration due to gravity - Gr Grashof number =gh³/v² - h electrode height - H cavity depth - k mass transfer coefficient - LD long dimension of expanded metal - R _h hydraulic radius - Sc Schmidt number=/D - SD small dimension of expanded metal - Sh Sherwood number=kh/D - void fraction - kinematic viscosity - density - electrode area per unit volume - electrode area per unit net area     

Mass transfer to planar and expanded metal electrodes in bubble columns

Mass transfer rates at planar electrodes and electrodes of expanded metal placed in the centre of a bubble column were measured. The gas velocity and the physical properties of the electrolytic solutions were varied and different types of expanded metal were investigated. In some cases increases in the mass transfer coefficient over the planar electrode value of more than 100% were obtained. Dimensionless correlations are presented for the different systems.Nomenclature A mean mesh aperture - D diffusivity - D _c column diameter - g acceleration due to gravity - Ga Galileo number =gL ³/v ² - Gr Grashof number =gL ³/v ² - k mass transfer coefficient - L electrode height - r radial position - R column radius - Re Reynolds number =R _h V _s/ - R _h hydraulic radius = / - Sc Schmidt number = /D - Sh Sherwood number =kL/D - V_s superficial velocity - gas void fraction - _M porosity of expanded metal - kinematic viscosity - density - electrode area per unit volume - electrode area per unit net area     

Evaluation of mass transport in copper and zinc electrodeposition using tracer methods

Control of the electrocrystallization process is essential in the deposition of metals from aqueous electrolytes. A knowledge of the influence of mass transfer on the metal ion reduction is a critical element in any number of electrolytic processes, particularly where relatively high current densities are desired. The use of more positive ion tracer techniques as a means of experimentally determining some of the mass transport properties of interest are described. Examples for copper, zinc and zinc alloys electrolysis are included.Nomenclature C _b concentration in the bulk of the solution - C _s concentration at the surface of the electrode - d hydraulic diameter of the cross section of the cell - D diffusion coefficient - e _Me equivalent weight of Me - F Faraday number - g acceleration due to gravity - Gr Grashof number - H hydrodynamic entrance length - (It) quantity of electricity (current times time) - J current density - J _dl diffusion limiting current density - k=J _dl/zFC mass transfer coefficient - L electrode length - P _Me deposited mass of Me - Re=vd/ Reynolds number - Sc=/D Schmidt number - Sh Sherwood number - v speed of electrolyte - z number of electrons exchanged in the electrode reaction - thickness of the diffusion layer - _d diffusion overvoltage - kinematic viscosity of electrolyte - average density across diffusion layer - _b bulk electrolyte density - ₁ density of the electrolyte at the surface of the electrode - rotation speed of the electrode     

Natural convection mass transfer at horizontal screens

The free convection mass transfer behaviour of horizontal screens has been investigated experimentally using an electrochemical technique involving the measurement of the limiting currents for the cathodic deposition of copper from acidified copper sulphate solutions. Screen diameter and copper sulphate concentration have been varied to provide a range ofSc.Gr from 22×10⁸ to 26×10¹⁰. Under these conditions, the data for a single screen are correlated by the equation:Sh=0.375(Sc.Gr)^0.305 Results have been compared with previous work on free convection at horizontal solid surfaces where mass transfer coefficients are somewhat lower.Mass transfer coefficients have been measured also for arrays of closely spaced parallel horizontal screens. The mass transfer coefficient was found to decrease with the number of screens forming the array.Symbols and units A area of mass transfer surface, cm² - C _b bulk concentration of ionic species, mol cm^–3 - D diffusivity, cm²s^–1 - F Faraday number, 96494 C g [equiv^–1] - Z number of electrons involved in the reaction - I _L limiting current, A - K mass transfer coefficient, cm s^–1 - Sh Sherwood number, dK/D - Sc Schmidt number,/D or/D - Gr Grashof numbergd ³/ ² _s - solution dynamic viscosity, g cm s^–1 - solution kinematic viscosity, cm² s^–1 - solution density, g cm^–3 - density difference between bulk solution and electrode/solution interface, g cm^–3 - _s solution density at electrode/solution interface, g cm^–3 - d screen diameter, cm - g gravitational acceleration, cm s^–2 On leave of absence, Chemical Engineering Department, Alexandria University, Alexandria, Egypt.     

Effect of gas evolution on mass transfer in an electrochemical reactor

Experiments were conducted to study the effect of gas bubbles generated at platinum microelectrodes, on mass transfer at a series of copper strip segmented electrodes strategically located on both sides of microelectrodes in a vertical parallel-plate reactor. Mass transfer was measured in the absence and presence of gas bubbles, without and with superimposed liquid flow. Mass transfer results were compared, wherever possible, with available correlations for similar conditions, and found to be in good agreement. Mass transfer was observed to depend on whether one or all copper strip electrodes were switched on, due to dissipation of the concentration boundary layer in the interelectrode gaps. Experimental data show that mass transfer was significantly enhanced in the vicinity of gas generating microelectrodes, when there was forced flow of electrolyte. The increase in mass transfer coefficient was as much as fivefold. Since similar enhancement did not occur with quiescent liquid, the enhanced mass transfer was probably caused by a complex interplay of gas bubbles and forced flow.List of symbols A electrode area (cm²) - a constant in the correlation (k = aRe ^m , cm s^–1) - C _{R, bulk} concentration of the reactant in the bulk (mol^–1 dm^–3) - D diffusion coefficient (cm² s^–1) - d _h hydraulic diameter of the reactor (cm) - F Faraday constant - Gr Grashof number =gL ³/² (dimensionless) - g gravitational acceleration (cm s^–2) - i _g gas current density (A cm^–2) - i _L mass transfer limiting current density (A cm^–2) - k mass transfer coefficient (cm s^–1) - L characteristic length (cm) - m exponent in correlations - n number of electrons involved in overall electrode reaction, dimensionless - Re Reynolds number =Ud _h^–1 (dimensionless) - Sc Schmidt number = D ^–1 (dimensionless) - Sh Sherwood number =kLD ^–1 (dimensionless) - U mean bulk velocity (cm s^–1) - x distance (cm) - _N equivalent Nernst diffusion layer thickness (cm) - kinematic viscosity (cm² s^–1) - density difference = (_L – ), (g cm^–3) - _L density of the liquid (g cm^–3) - average density of the two-phase mixture (g cm^–3) - void fraction (volumetric gas flow/gas and liquid flow)     

Effects of gas bubbles on the current and potential profiles within porous flow-through electrodes

This paper presents a mathematical model to calculate the distributions of currenti(x), potentialE(x), gas void fraction (x) and pore electrolyte resistivity (x) within porous flow-through electrodes producing hydrogen. It takes into consideration the following effects: (i) the kinetics of the interfacial charge transfer step, (ii) the effect of the non-uniformly generated gas bubbles on the resistivity of the gas-electrolyte dispersion within the pores of the electrode (x) and (iii) the convective transport of the electrolyte through the pores. These effects appear in the form of three dimensional groups i.e.K=i _o L where i_o is the exchange current density, is the specific surface area of the electrode andL its thickness.= ⁰ L where ⁰ is the pore electrolyte resistivity and =/Q where is a constant, =tortuosity/porosity of the porous electrode andQ is the superficial electrolyte volume flow rate within it. Two more dimensionless groups appear: i.e. the parameter of the ohmic effect =K/b and the kinetic-transport parameterI=K. The model equations were solved fori(x),E(x), (x) and (x) for various values of the above groups.Nomenclature specific surface area of the bed, area per unit volume (cm^–1) - b RT/F in volts, whereR is the gas constant,T is the absolute temperature (K) - B =[1–(I ² Z/4)], Equation 9a - C =(1–B ²), Equation 9b - E(L) potential at the exit face (V) - E(0) potential at the entry face (V) - E(x) potential at distancex within the electrode (V) - E _rev reversible potential of the electrochemical reaction (V) - F Faraday's constant, 96500 C eq^–1 - i _o exchange current density of the electrode reaction (A cm^–2 of true surface area) - i(L) current density at the exit face (A cm^–2 of geometrical cross-sectional area of the packed bed) - I K =i _oL(/Q) (dimensionless group), Equation 7d - K =i _oL, effective exchange current density of the packed bed (A cm^–2) Equation 7a - L bed thickness (cm) - q tortuosity factor (dimensionless) - Q superficial electrolyte volume flow rate (cm³ s^–1) - x =position in the electrode (cm) - Z =exp [(0)], Equation 7f - transfer coefficient, =0.5 - =K/b=(i ₀ L ⁰ L)/b (dimensionless group) Equation 7e - (x) gas void fraction atx (dimensionless) - = ⁰ L, effective resistivity of the bubble-free pore electrolyte for the entire thickness of the electrode ( cm²) - (0) polarization at the entry face (V) - (L) polarization at the exit face (V) - =q/, labyrinth factor - constant (cm³ C^–1), Equation 3a - =/Q (A ^–1) conversion factor, Equation 3b - porosity of the bed - (x) effective resistivity of the gas-electrolyte dispersion within the pores ( cm) - ⁰ effective resistivity of the bubble-free pore electrolyte ( cm)     

Electrochemical mass transfer studies in open cavities

Experimental measurements on free convection mass transfer in open cavities are described. The electrochemical deposition of copper at the inner surface of a cathodically polarized copper cylinder, open at one end and immersed in acidified copper sulphate was used to make the measurements. The effects on the rate of mass transfer of the concentration of the copper sulphate, the viscosity of the solution, the angle of orientation, and the dimensions of the cylinder were investigated. The data are presented as an empirical relation between the Sherwood number, the Rayleigh number, the Schmidt number, the angle of orientation and the ratio of the diameter to the depth of the cylinder. Comparison of the results with the available heat transfer data was not entirely satisfactory for a number of reasons that are discussed in the paper.Nomenclature C _b bulk concentration of Cu⁺⁺ (mol cm^–3) - C _b bulk concentration of H₂SO₄ (mol cm^–3) - C _o concentration of Cu⁺⁺ at cathode (mol cm^–3) - C _o concentration of H₂SO₄ at cathode (mol cm^–3) - D cavity diameter (cm) - D diffusivity of CuSO₄ (cm² s^–1) - D diffusivity of H₂SO₄ (cm² s^–1) - Gr Grashof number [dimensionless] (=Ra/Sc) - g acceleration due to gravity (=981 cm s^–2) - H cavity depth (cm) - h coefficient of heat transfer (Wm ^–2 K^–1) - i _L limiting current density (mA cm^–2) - K mass transfer coefficient (cm s^–1) - K ₁,K ₂ parameters in Equation 1 depending on the angle of orientation () of the cavity (see Table 3 for values) [dimensionless] - k thermal conductivity (W m^–1 K^–1) - L ^* characteristic dimension of the system (=D for cylindrical cavity) (cm) - m exponent on the Rayleigh number in Equation 1 (see Table 3 for values) [dimensionless] - Nu Nusselt number (=hL ^* k^–1) [dimensionless] - n exponent on the Schmidt number in Equation 1 (see Table 3 for values) [dimensionless] - Pr Prandtl number (=v/k) [dimensionless] - Ra Rayleigh number (defined in Equation 2) [dimensionless] - Sc Schmidt number (=v/D) [dimensionless] - Sh Sherwood number (=KD/D) [dimensionless] - ^t H⁺ transference number for H⁺ [dimensionless] - ^t Cu⁺⁺ transference number for Cu⁺⁺ [dimensionless] - specific densification coefficient for CuSO₄ [(1/)/C] (cm³ mol^–1) - specific densification coefficient for H₂SO₄ [(1/)/C] (cm³ mol^–1) - k thermal diffusivity (cm² s^–1) - dynamic viscosity of the electrolyte (g cm^–1 s^–1) - kinematic viscosity of the electrolyte (= /)(cm² s^–1) - density of the electrolyte (g cm^–3) - angle of orientation of the cavity measured between the axis of the cavity and gravitational vector (see Fig. 1) [degrees] - parameter of Hasegawaet al. [4] (=(2H/D))5/4 Pr^{– 1/2}) [dimensionless]     

Effect of drag-reducing polymer on the rate of mass transfer in fixed-bed reactors

Rates of mass transfer were measured for the cementation of copper from dilute copper sulphate solutions containing polyethylene oxide drag-reducing polymer on a fixed bed of zinc pellets. Starting from a Reynolds number (Re) of 550, the rate of mass transfer was found to decrease by an amount ranging from 7.5 to 51% depending onRe and polymer concentration. The percentage decrease in the rate of mass transfer increased with increasingRe, passed through a maximum atRe=1400 and then decreased rapidly with further increase inRe. The possibility of using drag-reducing polymers to reduce power consumption in fixed-bed operation was discussed in the light of the present and previous results.Nomenclature A cross-section of reactor (m²) - a specific area of bed (m²) - C copper sulphate concentration at timet (moll^–1) - C ₀ initial copper sulphate concentration (moll^–1) - D diffusivity of copper sulphate (m₂s^–1) - d _p particle diameter (m) - J _d mass transfer J-factor (StSc ^2/3) - K mass transfer coefficient (m s^–1) - L bed height (m) - Q volumetric flow rate (m³s^–1) - Re Reynolds number (V _i d _p/) - Sc Schmidt number (/_D) - St Stanton number (K/V _i) - V volume of copper sulphate solution (m³) - V _i interstitial velocity (V _s/), (ms^–1) - V _s superficial velocity (ms^–1) - bed porosity - solution viscosity (kg m s^–1) - solution density (kg m^–3) - storage tank residence time (s)     

Effect of gas evolution on current distribution and ohmic resistance in a vertical cell under forced convection conditions

The volume fraction of gas bubbles in a vertical cell with a separator was evaluated on the basis of the Bruggemann equation by taking into account the increase in velocity of the rising gas bubbles when fresh solution without gas bubbles is supplied to the bottom of the cell at constant velocity. This enhancement of the velocity results from an increase in the volume of gases evolving at the working electrode. The following three cases for overpotential at the working electrode were considered: no overpotential, overpotential of the linear type and of the Butler-Volmer type. The volume fraction, _h , at the top of the cell was expressed as a function of the dimensionless height of the cell and kinetic parameters. The total cell resistance can be expressed by {(2/5 _h )[1– _h )^–3/2–1+ _hD;]+µ}₁ d ₁/wh, where ₁ is the resistivity of the solution without gas bubbles,d ₁ the interelectrode distance,w the cell width,h the cell height and the parameter involving overpotential and resistance of the separator. It was found that there is an optimum value of the interelectrode distance. The optimum value is about a quarter of the value for the case of constant gas rise velocity, which corresponds to a closed system.Nomenclature b linear overpotential coefficient - C proportionality constant given by Equation 2 - d ₁ interelectrode distance - d ₂ thickness of the separator - F Faraday constant - h height of the cell - i current density - l total current - t ₀ exchange current density - k parameter given byd ₁(z)^1/2 - n number of electrons transferred - p gas pressure - r dimensionless cell resistance defined by Equation 16 - R gas constant - R _t total cell resistance - T temperature - u auxiliary function defined by Equation 37 - v solution velocity in the cell - v ₀ solution velocity at the bottom of the cell - v _h solution velocity at the top of the cell - V voltage at the working electrode - V _eq voltage at the working electrode when no current flows - w width of the electrode - y axis in the vertical direction from the bottom of the cell - z dimensionless variable fory, defined by Equation 8 - z _h dimensionless variable forh, defined by [C(V– V _eq/(₁ d ₁ v ₀]h - anodic transfer coefficient in the Butler-Volmer equation - volume fraction of gas bubbles in the cell - _h volume fraction of gas bubbles at the top of the cell - dimensionless cell voltage, given by Equation 38 - Butler-Volmer overpotential - Butler-Volmer overpotential when current density,I/wh, flows through the electrode, as described in Equation 42 - µ parameter representing either µ_S, µ_S + µ_L or µ_S + µ_BV - µ_BV ratio defined by Equation 41 - µ_L ratio defined byb/(₁ d ₁) - µ_S ratio defined by ₂ d ₂/₁ d ₁ - ₁ resistivity of the solution phase without gas bubbles - ₁(y) resistivity of the solution phase with gas bubbles at levely - ₂ resistivity of the separator - kinetic parameter in the Butler-Volmer equation, given by Equation 39     

Enhanced mass transfer to a rotating cylinder electrode with axial flow

A study of the rotating concentric cylindrical electrode has been made, in which the enhanced mass transfer rate by turbulence promotors to a smooth cylinder has been measured. When a special polypropene cloth was applied in the annulus an increase in the Sherwood number was detected, up to six times the value for a smooth cylinder at low Taylor numbers.Nomenclature A electrode area, dl (m²) - C ₀ bulk concentration (mol m^–3) - D diffusion coefficient (m² s^–1) - e annular gap,R-r (m) - F Faraday's constant, 96487 (As mol^–1) - I _l limiting current (A) - k _l mass transfer coefficient,I _l /nFC ₀ A (m s^–1) - l electrode height (m) - n number of electrons - r, R radius of inner and outer cylinder (m) - u axial liquid velocity (m s^–1) - angular velocity (rad s^–1) - kinematic viscosity (m² s^–1) - liquid density (kg m^–3) Dimensionless numbers Re _a axial Reynolds number 2eu/ - Re rotational Reynolds number 2r ²/ - Sc Schmidt number /D - Sh ^* rotational Sherwood number 2rk _l /D - Sh combined flow Sherwood number 2ek _l /D - St Stanton numberSh/Re /Sc - Ta Taylor number=re/(e/r)^1/2 - a, b, c power indices     

Natural convection mass transfer at spheres

Mass transfer by natural convection at spheres has been studied by an electrochemical technique involving limiting current measurement for the anodic dissolution of copper spheres in phosphoric acid. Acid concentration and sphere diameter were changed to provide values ofSc. Gr ranging from 2.85×10¹⁰to 2.15×10¹¹; under these conditions the mass transfer data was correlated by the equation:Sh=0.15 (Sc. Gr)^0.33 Nomenclature I Limiting current density - K mass transfer coefficient - F Faraday's constant - C Saturation solubility of copper phosphate in phosphoric acid - Z number of electrons involved in the reaction - Sh Sherwood number=Kd/D - Sc Schmidt number=v _ave/D - Gr Grashof number=gd ³ ( _i– _b)/v _ave ² _i - d Sphere diameter - D diffusivity of copper ions - u _b viscosity in the bulk liquid - u _i viscosity at the interface - _b density in the bulk liquid - _i density at the interface - g acceleration due to gravity - v _ave average Kinematic viscosity - Nu Nusselt number - Pr Prandtl number     

Current distribution in a two-dimensional narrow gap cell composed of a gas evolving electrode with an open part

On the basis of the observation of gas bubbles evolved by electrolysis, a two-dimensional vertical model cell composed of electrodes with open parts for releasing gas bubbles to the back side is proposed. The model cell consists of two layers. One layer forms a bubble curtain with a maximum volume fraction of gas bubbles in the vicinity of the working electrode with open parts. The other. being located out of the bubble layer, is a convection layer with a small volume fraction distributed in the vertical direction under forced convection conditions. The cell resistance and the current distribution were computed by the finite element method when resistivity in the back side varied in the vertical direction along the cell. The following three cases for overpotential were considered: no overpotential, overpotential of the linear type and overpotential of the Butler-Volmer type. It was found that the cell resistance was determined not only by the interelectrode gap but also by the percentage of open area and in some cases by the superficial surface area. The cell resistance varied only slightly with the distribution of the bubble layer in the back side.Nomenclature b linear overpotential coefficient given byb=/i - C proportionality constant given by Equation 15 - d ₁ distance between front side of working electrode and separator - d ₂ thickness of separator - F Faraday constant - I total current per half pitch - i current density at working electrode - i ₀ exchange current density - L length of a real electrolysis cell - n number of electrons transferred in electrode reaction - O _p percentage of open area given by Equation 1 - p pitch, i.e. twice the length of the unit cell, defined by 2(BC) in Fig. 4 - q thickness of bubble curtain, defined by (AM) in Fig. 4 - R gas constant - r _t total cell resistance - r unit-cell resistance defined by (V – V _eq)/I - r _rs residue ofr from sum ofr ₀ andr - r ₀ ohmic resistance of solution when0 _p=0 - r resistance due to overpotential when0 _p=0 - s electrode surface ratio or superficial surface area given by Equation 2 for the present model - T absolute temperature - t thickness of working electrode defined by EF in Fig. 4 - V cell voltage - V _eq open circuit potential difference between working and counter electrodes - solution velocity in cell - ₀ solution velocity at bottom of cell - w width of working electrode, defined by 2(DE) in Fig. 4 - x abscissa located on cell model - y ordinate located on cell model - anodic transfer coefficient - linear overpotential kinetic parameter defined byb/[_bc(p/2)] - d infinitesimally small length on the boundary - volume fraction of gas bubbles in cell - dimensionless cell voltage defined bynF(V – V _eq)/RT - overpotential at working electrode - Butler-Volmer overpotential kinetic parameter defined by [nFi _0bc(p/2)]/RT - coordinate perpendicular to boundary of model cell - ₁ resistivity of bubble-free solution - ₂ resistivity of separator - _bc resistivity of bubble curtain - potential in cell     

A parametric study of copper deposition in a fluidized bed electrode

The behaviour of a fluidized bed electrode of copper particles in an electrolyte of deoxygenated 5×10^–1 mol dm^–3Na₂SO₄–10^–3mol dm^–3H₂SO₄ containing low levels of Cu(II), is described as a function of applied potential, bed depth, flow rate, particle size range, Cu(II) concentration and temperature. The observed (cross sectional) current densities were more than two orders of magnitude greater than in the absence of the bed, and current efficiencies for copper deposition were typically 99%.No wholly mass transport limited currents were obtained, due to the range of overpotentials within the bed. The dependence of the cell current on the experimental variables (excluding temperature) was determined by regression analysis. The values of exponents for some of the variables are close to those expected, while others (for concentration and flow rate) reveal interactions between the experimental parameters. Nevertheless the values of the correlation coefficient matrix are low (except for the term relating expansion and flow rate), so that cross terms may be neglected in modelling the system at the first level of approximation.Nomenclature d mean particle diameter (mm) - E electrode potential, ( _m– _s)_r+(x) (V vs ref) wherer denotes the value of ( _m- _s) at the reversible potential - I (membrane) current density (A m^–2) - L static bed depth (mm) - M concentration of electroactive species (mol dm^–3) - T catholyte temperature (K) - u catholyte flow rate (mm s^–1) - x distance in the bed from the feeder electrode atx=0 - XL expanded bed depth (mm) - bed expansion (fraction of static bed depth) - _m metal phase potential (V) - _s solution phase potential (V) - _m metal phase resistivity (ohm m) - _s solution phase effective resistivity (ohm m) - overpotential (V)     

Dielectric properties of anodic oxide layers on tantalum

The dielectric properties of the anodically formed oxide layers on tantalum in contact with electrolyte were analysed by measuring the frequency and temperature dependence of the impedance. It has been found that the frequency dependence of the series capacitance and resistance component of the impedance in the audio frequency range are in accordance with Young's relation. In order to explain such behaviour the electrical resistivity is assumed to vary exponentially with distance through the oxide layer. This variation can be ascribed to the occurrence of the exponential change of oxygen vacancies in the anodic layer during the growth of the oxide layer. The activation energy was obtained from the temperature dependence of the series capacitance. In the paper the unsimplified Young's relations have been proved to be K-K transformable.Nomenclature C _s series capacitance (F) - R _s series resistance () - f frequency of applied signal (Hz) - x integration variable of frequency (Hz) - A area (cm²) - K characteristic length (cm^–1) - d oxide layer thickness (cm) - y distance through oxide (cm) - a _C slope of linear part of 1/C _s against logf plot (Equation 5) - a _R slope of linear part ofR _s against 1/f plot (Equation 3) - relative permittivity of oxide layer - (y) resistivity at distancey ( cm) - (0) resistivity on positiony=0 ( cm) - (d) resistivity on positiony=d ( cm) - T absolute temperature (K) - k Boltzmann constant (eV K^–1) - activation energy (eV) - z complex variable,z=x+iy, - Res residue     

Static and dynamic radii of polystyrene in ethylbenzene and tetrahydrofuran

Summary Hydrodynamic radii, R _H ^D and R _H , determined, respectively, from — translational diffusion coefficients and intrinsic viscosities are compared with radii of gyration S_Z for polystyrene of narrow molecular weight distribution in good-solvent systems, ethylbenzene and tetrahydrofuran. The S_Z data indicate THF is of comparable solvating power to ethylbenzene which, in turn, based on literature data is similar to benzene in its affinity for polystyrene. The ratios ₁ = S_z/R _H ^D and ₂=R _H /R _H ^D are much larger for polystyrene in ethylbenzene than in tetrahyaroruran. These results are interpreted to indicate the presence of a large draining effect in the ethylbenzene system.     

Application of metallic foams in an electrochemical pulsed flow reactor Part I: Mass transfer performance

This paper describes mass transfer in a porous percolated pulsated electrochemical reactor (E3P reactor), fitted with nickel foam electrodes in an axial configuration. The work is aimed at optimization of the mass transfer conditions in electroorganic reactions such as the oxidative cleavage of diols or the conversion of DAS (diacetone-l-sorbose) into DAG (diacetone-2-keto-l-gulonic acid). The use of nickel foam as an electrode material is of interest for these electrocatalytic reactions due to its high specific surface area (4000 to 11000 m^–1) and its high porosity (over 0.97). The electroreduction of ferricyanide has been chosen as a test reaction in order to correlate the mass transfer coefficient with the overall flow velocity and the amplitude and frequency of the electrolyte pulsation. Four foam grades have been tested.List of symbols a pulsation amplitude (m) - A _ve dynamic specific area of the foam: surface area per volume of material (m^–1) - C ferricyanide concentration in the cell (mol m^–3) - D diffusion coefficient of ferricyanide (m² s^–1 - d _m mean path of a particle in the threedimensional electrode (m) - d _R diameter of the reactor column (m) - d _p mean foam pore diameter of the foam (m) - e thickness of the electrode bed (m) - f pulsation frequency (Hz) - F Faraday number (C mol^–1) - I limiting diffusion current (A) - k _d mass transfer coefficient with pulsation (m s^–1) - k _o mass transfer coefficient without pulsation (m s^–1) - n number of electrons in the electrochemical reaction - Q _v volummetric flow rate through the reactor (m³ s^–1) - Re Reynolds number Re = U _o d _R v ^–1 - Re _pore Reynolds number based on mean pore diameter d _p, Re _pore = U ₀d _p^–1µ^–1 - S active surface area of the electrode (m²) - Sc Schmidt number, Sc = vD ^–1 - Sh Sherwood number, Sh = k _d d _R D ^–1 - Sh _pore Sherwood number based on mean pore diameter d _p, Sh _pore = k _d d _p D ^–1 - Sr Strouhal number, Sr = aU ₀ ^–1 - t _r mean residence time (s) - U ₀ permanent superficial velocity U ₀ = Q _v/(d _R ²/4) (ms^–1) Greek letters porosity of the foam - µ dynamic viscosity (kg m^–1 s^–1) - kinematic viscosity (m² s^–1) - liquid density (kg m^–3) - pulsation, = 2f (rad s^–1) - tortuosity of porous medium     

Diffusion-controlled current distributions near cell entries and corners

This paper reports experimental work undertaken to explore diffusion-controlled current distributions immediately downstream of sudden changes in flow cross-sectional area such as may occur at the entry to electrochemical flow cells. Nozzle flows expanding into an axisymmetric circular duct and into a square duct have been investigated using the reduction of ferricyanide ions on nickel micro-electrodes as the electrode process. The spanwise distribution of current has also been studied for the case of the square cell where secondary corner flows are significant.Nomenclature A electrode area (cm²) - c bulk concentration of transferring ions (mol dm^–3) - D cell diameter (cm) - D Diffusion coefficient (cm₂s^–1) - F Faraday number (96 486 C mol^–1) - I limiting electrolysis current (A) - k mass transfer coefficient (cm s^–1) - N nozzle diameter (cm) - u mean fluid velocity (cm s^–1) - x distance downstream from point of entry to cell (cm) - z number of electrons exchanged - electrolyte viscosity (g s^–1 cm^–1) - electrolyte density (g cm^–3) - (Re)_D duct Reynolds number,Du/ - (Re)_N nozzle Reynolds number,Nu/ - (Sc) Schmidt number,/D) - (Sh) Sherwood number,kD/D)     

Turbulent mass transfer in small radius ratio annuli

Mass transfer in annuli for both fully developed laminar and turbulent flow conditions has been studied with respect to available experimental data. It is shown that prediction of the Sherwood number for the inner annular wall based on the hypothesis of coincidence of the zero shear stress position for laminar and turbulent flows leads to serious error in the case of small radius ratio. Also it is shown that in contrast with plain tubes the curvature in small radius ratio annuli should be taken into account for the case of small Reynolds numbers. In consequence, the well-known Leveque equation can be used for the calculation of the mass transfer coefficient in annuli only under certain conditions. Possibilities of electrodiffusion diagnostics for the precise determination of the zero shear stress position in annuli are discussed.List of symbols A cross-section flow area (m²) - a =r ₁/r ₂ annular radius ratio (–) - mean fluctuation and bulk concentration (mol m^–3) - D molecular diffusivity (m²s^–1) - d _b hydraulic diameter (m) - f,f ₁,f ₂ overall, inner and outer wall friction factors (–) - f = ₁/ near wall velocity gradient (s^–1) - pressure drop per unit of length (Pam^–1) - K _L average mass transfer coefficient (ms^–1 ) - k =r ₀/r _0,L ratio of zero shear stress position in turbulent and laminar flows (–) - L mass transfer surface length (m) - L _D diffusion leading edge length (m) - L _ent diffusion entrance length (m) - P _W wetted perimeter (m) - Re =U _av d _h/ Reynolds number (–) - r radial distance from conduit axis (m) - r ₀,r _o,L radial distance of zero shear stress position in turbulent and laminar flows (m) - r ₁,r ₂ radius of inner and outer annular cylinders (m) - Sc = /D molecular Schmidt number (–) - Sh =K _L d _h/D Sherwood number (–) - U _av average liquid velocity (ms^–1) - u,u mean and fluctuation axial velocity (ms^–1) - , mean and fluctuation radial velocity (ms^–1) - y = r – r ₁ distance from the inner wall (m) - y = (/₁)^1/2 dynamic length (m) - Z distance in direction of the flow (m) Greek symbols _D diffusion layer thickness (m) - µ dynamic viscosity (Pa s) - kinematic viscosity (m²s^–1) - density (kgm^–3) - shear stress (Pa) - _W wall shear stress for tube and plate channel (Pa) - ₁, ₂ wall shear stress for inner and outer annular cylinders (Pa) - Geometrical factor with respect to k-function (–) - _R, _K geometrical factor with respect to Rothfus or Kays-Leung equations (–) - ratio of radial distance of zero shear stress position to outer radius in laminar flow (–)     

Behaviour of a tall vertical gas-evolving cell Part II: Distribution of current

Vertical electrolysers with a narrow electrode gap are used to produce gases, for example, chlorine, hydrogen and oxygen. The gas voidage in the solution increases with increasing height in the electrolyser and consequently the current density is expected to decrease with increasing height. Current distribution experiments were carried out in an undivided cell with two electrodes each consisting of 20 equal segments or with a segmented electrode and a one-plate electrode. It was found that for a bubbly flow the current density decreases linearly with increasing height in the cell. The current distribution factor increases with increasing average current density, decreasing volumetric flow rate of liquid and decreasing distance between the anode and the cathode. Moreover, it is concluded that the change in the electrode surface area remaining free of bubbles with increasing height has practically no effect on the current distribution factor.Notation A _e electrode surface area (m²) - A _e,s surface area of an electrode segment (m²) - A _{e, 1–19} total electrode surface area for the segments from 1 to 19 inclusive (m²) - A _e,a anode surface area (m²) - A _e,a,h A _e,a remaining free of bubbles (m²) - A _e,e cathode surface area (m²) - A _e,c,h A _e,c remaining free of bubbles (m²) - a ₁ parameter in Equation 7 (A^–1) - B current distribution factor - B _r B in reverse position of the cell - B _s B in standard position of cell - b _a Tafel slope for the anodic reaction (V) - b _c Tafel slope for the cathodic reaction (V) - d distance (m) - d _ac distance between the anode and the cathode (m) - d _wm distance between the working electrode and an imaginary membrane (m) (d _wm=0.5d _wt=0.5d _ac) - d _wt distance between the working and the counter electrode (m) - F Faraday constant (C mol^–1) - h height from the leading edge of the working electrode corresponding to height in the cell (m) - h _e distance from the bottom to the top of the working electrode (m) - I current (A) - I _s current for a segment (A) - I ₂₀ current for segment pair 20 (A) - I _1–19 total current for the segment pairs from 1 to 19 inclusive (A) - i current density (A m^–2) - i _av average current density of working electrode (A m^–2) - i _b current density at the bottom edge of the working electrode (A m^–2) - i ₀ exchange current density (A m^–2) - i _0,a i ₀ for anode reaction (A m^–2) - i _l current density at the top edge of the working electrode (A m^–2) - n ₁ parameter in Equation 15 - n _s number of a pair of segments of the segmented electrodes from their leading edges - Q _g volumetric rate of gas saturated with water vapour (m³ s^–1) - Q ₁ volumetric rate of liquid (m³ s^–1) - R resistance of solution () - R ₂₀ resistance of solution between the top segments of the working and the counter electrode () - R _p resistance of bubble-free solution () - R _p,20 R _p for segment pair 20 () - r _s reduced specific surface resistivity - r _s,0 r _s ath=0 - r _s,20 r _s for segment pair 20 - r _s, r _s for uniform distribution of bubbles between both the segments of a pair - r _s,,20 r _s, for segment pair 20 - T temperature (K) - U cell voltage (V) - U _r reversible cell voltage (V) - v ₁ linear velocity of liquid (m s^–1) - v _1,0 v ₁ through interelectrode gap at the leading edges of both electrodes (m s^–1) - x distance from the electrode surface (m) - gas volumetric flow ratio - ₂₀ at segment pair 20 - specific surface resistivity ( m²) - _t at top of electrode ( m²) - _p for bubble-free solution ( m²) - _b at bottom of electrode ( m²) - thickness of Nernst bubble layer (m) - ₀ ath=0 (m) - _{0,i 0} ati - voidage - _x,0 atx andh=0 - _0,0 voidage at the leading edge of electrode wherex=0 andh=0 - _0,0 ati _b - _0,0 ati=i _t - _,h voidage in bulk of solution at heighth - _,20 voidage in bubble of solution at the leading edge of segment pair 20 - _lim maximum value of _0,0 - overpotential (V) - _a anodic overpotential (V) - _c cathodic overpotential (V) - _h hyper overpotential (V) - _h,a anodic hyper overpotential (V) - _h,c cathodic hyper overpotential (V) - fraction of electrode surface area covered by of bubbles - _a for anode - _c for cathode - resistivity of solution ( m) - _p resistivity of bubble-free solution ( m)     

Generalized analytical expressions for Tafel slope,reaction order and a.c. impedance for the hydrogen evolution reaction (HER): mechanism of HER on platinum in alkaline media

Following the generally accepted mechanism of the HER involving the initial proton discharge step to form the adsorbed hydrogen intermediate, which is desorbed either chemically or electrochemically, generalized expressions for the Tafel slope, reaction order and the a.c. impedance for the hydrogen evolution reaction are derived using the steady-state approach, taking into account the forward and backward rates of the three constituent paths and the lateral interactions between the chemisorbed intermediates. Limiting relationships for the Tafel slope and the reaction order, previously published, are deduced from these general equations as special cases. These relationships, used to decipher the mechanistic aspects by examining the kinetic data for the HER on platinum in alkaline media, showed that the experimental observations can be consistently rationalized by the discharge-electrochemical desorption mechanism, the rate of the discharge step being retarded on inactive platinum compared to the same on active platinum.Nomenclature C _d double-layer capacity (µF cm^–2) - E _rev reversible electrode potential (V) - F Faraday number (96 487 C mol^–1 ) - R gas constant - T temperature (K) - Y _f Faradaic admittance (^–1 cm^–2) - Y _t Total admittance (^–1 cm^–2) - Z _f Faradaic impedance ( cm²) - i _f total current density (A cm^–2) - i _nf nonfaradaic current density (A cm^–2) - j - k ₀ ¹ rate constant of the steps described in Equations 1 to 3 (mol cm^–2 s^–1 ) - j - qmax saturation charge (µC cm^–2) - Laplace transformed expressions for i, and E - _{1 3} symmetry factors for the Equations 1 and 3 - saturation value of adsorbed intermediates (mol cm^–2) - overpotential - coverage by adsorbed intermediates - angular frequency This paper is dedicated to Professor Brian E. Conway on the occasion of his 65th birthday, and in recognition of his outstanding contribution to electrochemistry.