Mass transfer at rotating cone electrodes

Upright and inverted rotating cone electrodes (apex half angle 52°) have been studied using electrodeposition of copper from an acid electrolyte (Sc=1770) by means of limiting current mass transport techniques. The behaviour suggests that it is appropriate to regard these electrodes as modified disc electrodes. The upright rotating cone electrode exhibits a flow transition atRe=1×10⁵. In laminar flowSh=4.5Re ^0.48 and in turbulent flowSh=0.04Re ^0.95. The inverted rotating cone electrode exhibits a flow transition atRe=6×10⁴. In laminar flowSh=4.5Re ^0.45 and in turbulent flowSh=0.04Re ^0.88. The data have been interpreted in terms of a coating thickness affected by throwing power effects and the use of a conical cathode cell for control of high speed electrodeposition processes is indicated.Nomenclature A electrode area (cm²) - C _b concentration (of bulk solution) (mol cm^–3) - c concentration difference (mol cm^–3) - D diffusion coefficient (cm² s^–1) - f/2 friction factor - F Faraday's constant (96485 A s mol^–1) - i _L limiting current density (A cm^–2) - J mass transfer flux (mol s^–1 cm^–2) - K _L mass transfer coefficient (cm s^–1) - l slant height of the cone (cm) - U peripheral velocity (cm s^–1) - x local condition coordinate (cm) - z no. of electrons - apex half-angle of cone - angular velocity (rad s^–1) - wall shear stress (dyn cm^–2) - kinematic viscosity (cm² s^–1) - Re Reynolds number=Ud/v - Sc Schmidt number=v/D - Sh Sherwood number=K _L d/D     

Modelling of gas-evolving electrolysis cells. II. Investigation of the flow field around gas-evolving electrodes

The flow field in front of and around hydrogen- or oxygen-evolving electrodes of different shapes has been investigated by Laser-Doppler anemometry. A strong influence of geometrical parameters on the structure of the flow field has been found. The vertical velocity component in front of a plane electrode decreases with distance. Due to the resulting pressure gradient a well-defined bubble curtain is formed at such electrodes. Gas voidage data derived from experimental velocity data are in close agreement with the predictions of the coalescence barrier model which is valid for electrolyte solutions.Nomenclature f frequency (s^–1) - F Faraday number (96487 As mol^–1) - G volumetric gas flow rate (cm³ s^–1) - h height (cm) - i current density (A cm^–2) - L volumetric liquid flow rate (cm³ s^–1) - N number of data points (1) - p pressure (Pa) - Q _t total volumetric flow rate (cm³ s^–1) - R _g gas constant (8.3144 J K^–1 mol^–1) - T temperature (K) - T _u degree of turbulence (1) - u linear flow velocity (cm s^–1) - u ⁰ superficial flow velocity (cm s^–1) - u _sw swarm velocity (cm s^–1) - x thickness (cm) - y depth (cm) Greek symbols _g gas voidage (1) - _m maximum gas voidage (1) - _e electron number (1) - mass density (g cm^–3) Paper presented at the 2nd International Symposium on Electrolytic Bubbles organized jointly by the Electrochemical Technology Group of the Society of Chemical Industry and the Electrochemistry Group of the Royal Society of Chemistry and held at Imperial College, London, 31st May and 1st June 1988.     

Criterion of completeness of electrolysis at flow porous electrodes

An equation is presented which allows the calculation of the critical solution flow velocity corresponding to complete reaction controlled by diffusion at flow porous electrodes. The equation has been experimentally confirmed with good accuracy for the mass transport-controlled reaction of the reduction of K₃Fe(CN)₆ at flow porous electrodes composed of fine platinum screens and of gilded graphite granules described in the literature. If the critical flow velocity can be determined experimentally, the equation may be used for the determination of the specific surface of the electrode or the diffusion coefficient of the process. In this way the specific surfaces of graphite electrodes have been determined, which also enabled the calculation of mass transfer coefficients and dimensionless correlations for the Sherwood Number andj _D-factor.List of symbols A/t' Empirical constant in Equation 5 - B Empirical constant in Equation 5 - d _p Particle diameter (cm) - D Diffusion coefficient (cm²s^–1) - j _D j _D -factor,j _D =(Sh)(Re)^–1(Sc)^–1/3 - k Coefficient of mass transfer (cm s^–1) - L Electrode height (cm) - M log₁₀e=0.4343 - r Pore radius (cm) - r ₁ Coefficient of correlation - R Limiting degree of conversion - R _c Critical limiting degree of conversion - R _c Average critical limiting degree of conversion - (Re) Reynolds Number, (Re)=ud _p / - R _h Hydraulic radius (cm) - s Specific surface (cm^–1) - (Sc) Schmidt Number, (Sc)=/D - Average Schmidt Number - (Sh) Sherwood Number, (Sh)=kd _p /D - T Absolute temperature (K) - u Superficial flow velocity (cm s^–1) - u _c Critical superficial flow velocity (cm s^–1) - w Interstitial flow velocity (cm s^–1) - Void fraction - Dynamic viscosity (poise) - Dimensionless parameter - Kinematic viscosity (cm²s^–1)     

Characterization of reticulate,three — dimensional electrodes

A study has been made of the mass transfer characteristics of a reticulate, three-dimensional electrode, obtained by metallization of polyurethane foams. The assumed chemical model has been copper deposition from diluted solutions in 1 M H₂SO₄. Preliminary investigations of the performances of this electrode, assembled in a filter-press type cell, have given interesting results: with 0.01 M CuSO₄ solutions the current density is 85 mA cm^–2 when the flow rate is 14 cm s^–1.List of symbols a area for unit volume (cm^–1) - C copper concentration (mM cm^–3) - c _L copper concentration in cathode effluent (mM cm^–3) - c ₀ copper concentration of feed (mM cm^–3) - C ₀ ⁰ initial copper concentration of feed (mM cm^–3) - d pore diameter (cm) - D diffusion coefficient (cm²s^–1) - F Faraday's constant (mcoul me _q ^–1 ) - i electrolytic current density on diaphragm area basis (mA cm^–2) - I overall current (mA) - K _m mass transfer coefficient (cm s^–1) - n number of electrons transferred in electrode reaction (me_q mM^–1) - P ] volumetric flux (cm³s^–1) - Q total volume of solution (cm³) - (Re) Reynold's number - S section of electrode normal to the flux (cm²) - (Sc) Schmidt's number - (Sh) Sherwood's number - t time - T temperature - u linear velocity of solution (cm s^–1) - V volume of electrode (cm³) - divergence operator - void fraction - u/K _m a(cm) - electrical specific conductivity of electrolyte (^–1 cm^–1) - _S potential of the solution (mV) - density of the solution (g cm^–3) - v kinematic viscosity (cm²s^–1)     

A comparative study of the effect of different drag-reducing polymers on electrochemical mass transfer

The effect of Polyox, Separan and CMC drag-reducing polymers on the rate of electrochemical mass transfer was studied using the cathodic reduction of K₃Fe(CN)₆ in neutral media at a rotating cylinder cathode. Reynolds number and polymer concentration were varied over the ranges 764–10470 and 10–200 ppm respectively. Under these conditions it was found that the three polymers reduce the rate of mass transfer by a maximum of 47%, 30%, and 17% for Polyox, Separan and CMC, respectively. Mass transfer data in the three polymer solutions was correlated by the following equations: for Polyox: (St)=0.051(Re)^–0.3 (Sc)^–0.644 (u/u ₀ ^–0.7 for Separan: (St)=0.065(Re)^–0.3 (Sc)^–0.644 (u/u ₀)^–0.7 for CMC: (St)= 0.075(Re)^–0.3 (Sc)^–0.644) (u/u ₀)^–0.5 List of symbols I limiting current density (A cm^–2) - Z number of electrons involved in the reaction - F Faraday's constant - K mass transfer coefficient (cm s^–1) - V linear velocity of the cylinder (cm s^–1) - D diffusion coefficient (cm²s^–1) - v kinematic viscosity (cm²s^–1) - d diameter of the cylinder (cm) - u, u ₀ viscosity of solutions with and without polymer respectively (P) - density (g cm^–3) - c concentration of Fe(CN) ₆ ^3– (mol cm^–3) - (St) Stantonnumber=K/V - (Sc) Schmidt number=v/D - (Re) Reynolds number=Vd/u     

Mass transfer at carbon fibre electrodes

Mass transfer at carbon fibre electrodes has been studied using the mass transfer controlled reduction of potassium hexacyanoferrate(III) to potassium hexacyanoferrate(II). Different geometrical configurations have been assessed in a flow-by mode, namely bundles of loose fibres with liquid flow parallel to the fibres, carbon cloth with flow parallel to the cloth and carbon felt with liquid flow through the felt. For comparison, mass transfer rates at a single fibre have been measured; the experimental data fit the correlationSh=7Re ^0.4. The same correlation can be used as a first approximation for felts. Mass transfer for fibre bundles and cloth under comparable conditions is much lower owing to channelling.Nomenclature c reactant concentration (mol m^–3) - c ₀ reactant concentration atx=0 (mol m^–3) - c _L reactant concentration atx=L (mol m^–3) - d fibre diameter (m) - D diffusion coefficient (m² s^–1) - F Faraday number (96 487 C) - h depth of the electrode (m) - i current density (A m^–2) - I current (A) - k mass transfer coefficient (m s^–1) - L length of the electrode (m) - n number of electrons - S specific surface area (m² m^–3) - u (superficial) velocity (m s^–1) - V _R reactor volume (m³) - w width of electrode (m) - x distance in flow direction (m) - current efficiency - electrode efficiency - characteristic length (m) - v kinematic viscosity (m² s^–1) - _s ⁿ normalized space velocity (m³ m^–3s^–1) - Re Reynolds number (ud/v) - Sh Sherwood number (kd/D) - Sc Schmidt number (v/D)     

Behaviour of a carbon felt flow by electrodes Part I: Mass transfer characteristics

The properties of a carbon felt electrode have been experimentally investigated with special attention to its possible application in the electrochemical recovery of heavy metals. The mass transfer process has been studied by means of the reduction of ferricyanide and cupric ions for a flow-by electrode operating under limiting current conditions. An empirical correlation between the Sherwood and Reynolds numbers has been used to compare the experimental data with those obtained by other authors for different porous electrodes.Notation a specific electrode area (m^–1) - a _v area per unit solid volume (m^–1) - C _in entering concentration of reacting species (kmol m^–3) - C _out exit concentration of reacting species (kmol m^–3) - d _f fibre diameter (m) - d _b hydraulic diameter of the felt fibres (m) - D diffusion coefficient (m² s^–1) - F Faraday number 96 487 (C mol^–1) - k _m mass transfer coefficient (m s^–1) - l_lim limiting current (A) - l length of the electrode (m) - L thickness of the electrode (m) - Q _L catholyte flow rate (m³ s^–1) - Re Reynolds numberRe=d _h u/v - Sh Sherwood numberSh=k _m d _h/D - u solution velocity in the empty cross-section (m s^–1) - X reaction conversion - z number of electrons in the electrochemical reaction Greek letters porosity of the felt - kinematic viscosity of the solution (m² s^–1) - _Rgq_A true and apparent density of the felt (kg m^–3)     

Investigation of hydrodynamic test systems for the selection of high flow rate resistant materials

Flow-dependent corrosion phenomena can be studied in the laboratory and on a pilot plant scale by a number of methods, of which the rotating disc, the rotating cylinder, the coaxial cylinder and the tubular flow test are the most important. These methods are discussed with regard to mass transfer characteristics and their applicability to flow-dependent corrosion processes and erosion corrosion. To exemplify the application of such methods to materials selection for seawater pumps, corrosion data of non-alloyed and low alloy cast iron are presented.Nomenclature (Sh) Sherwood number - (Re) Reynolds number - n exponential of Reynolds number - shear stress (Pa) - dynamic viscosity (Pa s) - du/dy velocity gradient (s^–1) - mass density (kg m^–3) - f friction factor - (Sc) Schmidt number - i _cor,i _c corrosion current density (mA cm^–2) - i _lim limiting current density (mA cm^–2) - u _cor corrosion rate (mm y^–1 or g m^–2d^–1) - u flow rate (ms^–1) - k constant - u _ph phase boundary rate (gm^–2d^–1) - z number of electrons exchanged - F Faraday number (96 487 As mol^–1) - D diffusion coefficient (m²s^–1) - c concentration (kmol m^–3) - L characteristic length (m) - kinematic viscosity (m² s^–1) - h gap width (m) - v volume rate (m³s^–1) - m rotation rate (min^–1) - u _rel relative rate of co-axial cylinders (m s^–1) - _H electrode potential versus SHE (V)     

Enhanced mass transfer using roughened rotating cylinder electrodes in turbulent flow

Cylindrical electrodes have been roughened by machining groove patterns, pyramidal knurling, and superimposing wires and meshes for which the degree of roughness has been calculated. By rotating the electrodes in a turbulent regime, mass transfer for cathodic copper electrodeposition has been measured and the degree of consequent enhancement (relative to an equivalent smooth cylinder) calculated. Typically, the surface area has been increased by 10–40% and the mass transfer rate by 100–300% for turbulent flow defined by 7000<Re<80 000.Nomenclature A (A _R) area of cathodic (rough) cylinder (cm²) - C exponent - C _B metal ion concentration in bulk solution (mol cm^–3) - d _s (d _R) diameter of smooth (rough) cylinders (cm) - D diffusion coefficient of metal ion (cm²s^–1) - F Faraday's constant - I _L limiting current density (mA cm^–2) - j ₀ dimensionless mass transfer factor (=ShSc ^c) - k _L mass transfer coefficient (=I _L/zFC _B) - k _s,k _R k _L values for smooth and rough cylinders - m, n exponents - P pitch, or roughness element spacing (cm) - Re Reynolds number (=Ud/v) - Ré d _R U _R/v - Re _s dU _s/v - Sc Schmidt number (=v/D) - Sh Sherwood number (=k _L d/D) - St Stanton number (=k _L/U) - U _s (U _R) peripheral velocity at smooth (rough) cylinders (cm s^–1) - U ₀ friction velocity (cm s^–1) - w width of wire mesh opening (cm) - z valency change, number of electrons - groove depth (cm) - kinematic viscosity (cm² s^–1) - groove width     

Effect of drag reducing polymers on the rate of mass transfer in relation to their use as corrosion inhibitors in pipelines under turbulent flow conditions

Rates of mass transfer between a turbulently flowing fluid containing CMC drag reducing polymer and the wall of a tube were measured in the mass transfer entry region using the electrochemical technique. Variables studied were polymer concentration, surface roughness and solution flow rate. Carboxymethyl cellulose (CMC) was found to reduce the mass transfer coefficient by an amount ranging from 15 to 37% depending on the operating conditions. The percentage decrease in the mass transfer coefficient becomes greater with increasing CMC concentration and Reynolds number. CMC was found to reduce the rate of mass transfer at rough surfaces (e ⁺>3) by an amount higher than that at a smooth surface. The possibility of using large polymers as drag reducers and corrosion inhibitors simultaneously in pipelines is indicated.Nomenclature I limiting current (A) - Z number of electrons involved in the reaction - F Faraday's constant - A projected (geometrical) area of the cathode (cm²) - K mass transfer coefficient (cm s^–1) - C concentration of ferricyanide ion (mole cm^–3) - e roughness height (cm) - d tube diameter (cm) - L length of transfer surface (cm) - St Stanton number (K/V) - Re Reynolds number (Vd/u) - Sc Schmidt number (v/D) - e ⁺ dimensionless height (eu ^*/v) - u ^* friction velocity [V(f/2)^1/2] (cm s^–1) - V solution velocity (cm s^–1) - f friction factor - v kinematic viscosity (cm² s^–1) - u viscosity (poise) - density (g cm^–3) - D diffusivity (cm²s^–1)     

The role of mass transfer in the electrolytic reduction of hexavalent chromium at gas evolving rotating cylinder electrodes

The rate of electrolytic reduction of hexavalent chromium from acidic solution at a hydrogen-evolving rotating cylinder lead cathode was studied under conditions of different current densities, Cr⁶⁺ concentrations and rotation speeds. The rate of the reaction was found to follow a first order rate equation. The specific reaction rate constant was found to increase with increasing rotation speed until a limiting value was reached with further increase in rotation speed. Mechanistic study of the reaction has shown that at relatively low rotation speeds the reduction of Cr⁶⁺ is partially diffusion controlled, at higher speeds the reaction becomes chemically controlled. The limiting specific reaction rate constant was related to the operating current density by the equationK=0.044i ^1.385. The current efficiency of Cr⁶⁺-reduction was measured as a function of current density, initial Cr⁶⁺ concentration and rotation speed. Possible practical applications are discussed.Nomenclature A electrode area (cm²) - a, b constants in Equations 5 and 13, respectively - C bulk concentration of Cr⁶⁺ at timet(M) - C _o initial concentration of Cr⁶⁺ (M) - C _i interfacial concentration of Cr⁶⁺ (M) - d cylinder diameter (cm) - D diffusivity of Cr⁶⁺ (cm² s^–1) - e _o standard electrode potential (V) - F Faraday's constant (96 487 C) - current consumed in hydrogen discharge (A) - i current density (A cm^–2) - I cell current (A) - K _l mass transfer coefficient (cm s^–1) - K _r mass transfer coefficient due to cylinder rotation (cm s^–1) - K _o natural convection mass transfer coefficient (cm s^–1) - K _g mass transfer coefficient due to hydrogen stirring (cm s^–1) - K ₂ specific reaction rate constant (cm s^–1) - K overall rate constant (cm s^–1) - m theoretical amount of Cr⁶⁺ reduced during electrolysis (g) - P gas pressure (atm) - R gas constant (atm cm³ mol^–1 K^–1) - T temperature (K) - t time (s) - V linear speed of the rotating cylinder (cm s^–1) - hydrogen discharge rate (cm³ cm^–2 s^–1) - V _s solution volume (cm³) - z electrochemical equivalent (g C^–1) - Z number of electrons involved in the reaction - Re Reynolds number (Vd/v) - Sh Sherwood number (K _r d/D) - Sc Schmidt number (v/D) - rotation speed (r.p.m.) - kinematic viscosity (cm² s^–1)     

The characterization of three-dimensional electrodes using an electrochemical tracer method

A new approach is suggested for the characterization of electrochemical reactors and is applied to three-dimensional electrodes. This approach permits the investigation of the fluid flow pattern through heterogeneous media and the overall reactivity of the bed. The fluid flow patterns have been derived by adapting the tracer method (well-known in chemical reaction engineering) for measurements on electrochemical reactors: auxiliary electrodes have been used both for the production and detection of concentration pulses. Experiments have been carried out on beds of glass beads, the size of the beads, height of the beds and flow rates being varied. The results are expressed as (Pe)-(Re) relationships. The reactivity of the beds has been determined using a new method, the mathematical background of which is due to be published. This method has been tested on electrochemically active beds of glass beads coated with copper and silver, the particle size and flow rates again being varied. The results are expressed ask=Sk _m(=SD/) relationships.List of symbols C concentration (mol cm^–3) - ¯D dispersion coefficient (cm² s^–1) - D diffusion coefficient (cm²s^–1) - diffusion layer thickness (cm) - d _p particle diameter (cm) - I(t) function defined by Equation 5 - K overall reactivity constant of the bed (s^–1) - k _m mass transfer coefficient (cm s^–1) - l distance along the length of the electrode (cm) - M _{1, 2} first and second moment of the distribution of residence times - fluid viscosity (g s^–1 cm^–1) - (Pe) Peclét number=UL/D - r electrochemical reaction rate (mol cm^–3 s^–1) - (Re) Reynolds number=Ud_p/. - fluid density (g cm^–3) - S specific surface area of the electrode (total surface/total volume) (cm^–1) - t time (s) - average residence time of the species entering the electrode (s) - U interstitial fluid velocity (cm s^–1) - v volumetric flow rate (cm³ s^–1) - free volume (cm³) - X the degree of a conversion - y ₁ (t) response of the three-dimensional electrode when the current is switched off - y ₂ (t) response of the three-dimensional electrode in the limiting current regime     

Effect of drag-reducing additives on the rate of mass transfer in a parallel-plate electrochemical flow reactor

The effect of polyox and CMC drag-reducing polymers on the rate of mass transfer in a parallel-plate flow cell was studied by measuring the limiting current for the cathodic reduction of potassium ferricyanide in alkaline medium. Reynolds number and polymer concentration were varied over the range 3500–21 000 and 10–200 ppm respectively. Under these conditions it was found that polyox and CMC reduce the rate of mass transfer by a maximum of 42% and 35% respectively.Nomenclature a a constant - C concentration of ferricyanide ion (g mol cm^–3) - D diffusivity of ferricyanide ion (cm²s^–1) - d _e equivalent diameter of the cell (4 x cross-sectional area/wetted perimeter) - F Faraday's constant (96 487 C mol^–1) - I limiting current density (A cm^–2) - K mass transfer coefficient (cm s^–1) - L electrode height (cm) - (Re) Reynolds number (d _e /u) - (Sc) Schmidt number (u/D) - (Sh) Sherwood number (Kd _e/D) - u solution viscosity (poise) - flow rate of the solution (cm s^–1) - Z number of electrons involved in the reaction - solution density (g cm^–3)     

Performance of bipolar trickle reactors

The performance of the bipolar trickle reactor has been studied using the electrochemical tracer technique. The theoretical equations for a semi-infinite dispersion model have been fitted to the experimental responses for the reactor with and without electrochemical reaction. Hydrodynamic parameters and reaction rate constants for copper deposition as functions of both the film Reynolds number and the dimensions of the bipolar trickle reactor have been derived and are interpreted in this paper.List of Symbols (Bo) Bodenstein number (uL _p/D) - C amplitude of the response curve (dimen sionless) - C ⁰ area under the response curve (mol cm^–3 s) - D dispersion coefficient (cm²s^–1) - h film thickness (cm) - k/h first order reaction rate constant (s^–1) - L length of the reactor (cm) - L _p length of the ring (cm) - n _r number of rings in a single layer - (Pe) Peclét number (uL/D) - (Re)_f film Reynolds number - r _i,r _o inner and outer radii of the ring (cm) - t time (s) - u mean liquid velocity (cm s^–1) - v volumetric liquid velocity (cm³ s^–1) - residence time (s) - kinematic viscosity (cm²s^–1)     

Mass transfer at rotating finned cylinders

Rates of mass transfer at rotating finned cylinders were studied by an electrochemical technique involving the measurement of the limiting current for the cathodic reduction of potassiun ferricyanide in a large excess of sodium hydroxide. The variables studied were fin height and Reynolds number. The ratio of the fin height to the cylinder diameter (e/d) ranged from 0·0185 to 0·075 while the Reynolds number ranged from 1047 to 10 470. Under these conditions, the mass transfer data could be correlated by the equationJ=0·714(Re)^–0.39(e/d)^0.2 Nomenclature L _L limiting current (A) - K mass transfer coefficient (cm s^–1) - Z number of electrons involved in the reaction - C ferricyanide concentration (moles cm^–3) - F Faraday's constant - A projected cathode area (cm²) - u dynamic viscosity (g cm^–1 s^–1) - density (g cm^–3) - V peripheral velocity at the rotating cylinder (cm s^–1) - D diffusion coefficient of ferricyanide ion (cm²s^–1) - d cylinder diameter (cm) - e fin height (cm) - J (St)(Sc)^0.664 ColburnJ factor - (Sc) u/(D) Schmidt number - (Re) Vd/u Reynolds number - (St) K/V Stanton number     

Optimal design of packed bed cells for high conversion

In connection with the electrochemical purification of metal containing waste waters, the realization of a high concentration decrease per pass is one of the goals of design optimization. For a packed bed cell with crossed current and electrolyte flow directions high conversion in conjunction with a large space time yield requires limiting current conditions for the whole electrode. For establishing the concentration profiles in the direction of flow a plug flow model is used. These considerations result in a new packed bed electrode geometry for which an analytical bed depth function is derived. The basic engineering equations of such packed bed electrodes are given, and design equations for different arrangements are developed. The reliability of this scaling-up method is shown by comparison of theoretically predicted and experimental performance data of two cells. Engineering aspects such as easy matching of cells to waste water properties and parametric sensitivity are discussed. Some technical applications are reported.Nomenclature and constants used in the calculations A _s specific electrode surface (cm^–1) - b(y) width of the packed bed (cm) - c(y) metal concentration (mol cm^–3) - C _e ^t total equivalent concentration of electroactive species (mol cm^–3) - D diffusion coefficient (cm² s^–1) - D _c conversion degree (1) - d _p(y) diameter of packed bed particles (cm) - F Faraday number (96.487 As mol^–1) - h(y) bed depth parallel to current flow direction (cm) - i() current density (A cm^–2) - i _b bed current density (A cm^–2) - i _g[c(y)] diffusion limited current density (A cm^–2) - mean current density of metal deposition (A cm^–2) - k(y) mass transfer coefficient (cm s^–1) - k 0.8121×10^–3 cms^–1/2 - U cell voltage (V) - u(y) flow velocity (cm s^–1) - v voidage (0.56) - v _A volume of anode compartement (cm³) - V _B volume of packed bed electrode (cm³) - v _D volume flow rate (cm³ s^–1) - W water parameter (mol cm^–2 A^–1) - x coordinate parallel to current flow (cm) - y coordinate parallel to electrolyte flow (cm) - y _ST ^E space time yield of the electrode (s^–1 or m³h^–1l^–1) - y _ST ^C space time yield of the cell (s^–1 or m³h^–1l^–1) - z coordinate normal to current and electrolyte flow (cm) - z _i charge number (1) - current efficiency (1) - ₁ overpotential near the feeder electrode (V) - ₂ overpotential near the membrane (V) - ₂- ₁ (V) - (x, y) overpotential at point (x, y) (V) - _s particle potential (V) - _s electrolyte potential (V) - X electrolyte conductivity (S cm^–1) - X _p particle conductivity (S cm^–1) - _s electrolyte conductivity (S cm^–1) - v kinematic viscosity (cm² s^–1) - slope of the feeder electrode (1)     

Mass transfer at rough gas-sparged electrodes

Rates of mass transfer were measured by the limiting current technique at a smooth and rough inner surface of an annular gas sparged cell in the bubbly regime. Roughness was created by cutting 55°V-threads in the electrode normal to the flow. Mass transfer data at the smooth surface were correlated according to the expression j = 0.126(Fr Re)^–0.226 Surface roughness of peak to valley height ranging from 0.25 to 1.5 mm was found to have a negligible effect on the mass transfer coefficient calculated using the true electrode area. The presence of surface active agent (triton) in the solution was found to decrease the mass transfer coefficient by an amount ranging from 5% to 30% depending on triton concentration and superficial air velocity. The reduction in the mass transfer coefficient increased with surfactant concentration and decreased with increasing superficial gas velocity.Nomenclature a constant - A electrode area (cm²) - C _p specific heat capacity Jg^–1 (K^–1) - C ferricyanide concentration (m) - d _c annulus equivalent diameter, (d _o – d _i) (cm) - d _o outer annulus diameter (cm) - d _i inner annulus diameter (cm) - D diffusivity of ferricyanide (cm²s^–1) - e peak-to-valley height of the roughness elements (cm) - e ⁺ dimensionless roughness height (eu ^*/) - f friction coefficient - F Faraday constant (96 500 Cmol^–1) - g acceleration due to gravity (cm s^–2) - h heat transfer coefficient (J cm^–2 s K) - I _L limiting current (A) - K mass transfer coefficient (cm s^–1) - K thermal conductivity (W cm^–1 K^–1) - V _g superficial air velocity (cm s^–1) - Z number of electrons involved in the reaction - Re Reynolds number (_L V _g d _e/) - J mass or heat transfer J factor (St Sc ^0.66) or (St Pr ^0.66), respectively - St Stanton number (K/V _g for mass transfer and h/C _p V _g for heat transfer) - Fr Froude number (V _g ² /d _e g) - Sc Schmidt number (/D) - Pr Prandtl number (C _p/K) - PL solution density (g cm^–3) - kinematic viscosity (cm²s^–1) - gas holdup - u ^* friction velocity = V _L(f/2) - diffusion layer thickness (cm) - solution viscosity (gcm^–1 s^–1)     

Effect of drag-reducing additives on the rate of mass transfer in diffusion-controlled electrochemical processes

Mass transfer between a rotating cylinder and a solution containing sodium carboxymethyl cellulose polymer, was studied using an electrochemical technique involving the reduction of potassium ferricyanide in a large excess of sodium hydroxide. The Reynolds number and polymer concentration were varied over the ranges 4100–41 000 and 10–500 ppm, respectively. Under these conditions, it was found that polymer addition reduces the mass transfer coefficient by 10–22% depending on Reynolds number and polymer concentration. The mass transfer data in polymer-containing solutions were found to fit the equation (St) = 0.07(Re)^–0.3(Sc)^–0.644.List of symbols I _L limiting current density (A cm^–2) - Z number of electrons involved in the reaction - F Faraday's constant (96 500 C) - K mass transfer coefficient (cm s^–1) - V linear velocity of the cylinder (cm s^–1) - angular velocity (rad s^–1) - D diffusion coefficient (cm² s^–1) - kinematic viscosity (cm² s^–1) - d diameter of the cylinder (cm) - u viscosity of the solution (poise) - density of the solution (g cm^–3) - C concentration (mol cm^–3) - (St) K/V, Stanton number - (Sc) /D, Schmidt number - (Re) d/u, Reynolds number     

Effect of anodically evolved gas bubbles on the rate of cathodic mass transfer in a vertical cylinder cell

Mass transfer coefficients were measured for the deposition of a copper from acidified copper sulphate solution at a vertical cylinder cathode stirred by oxygen evolved at a coaxial vertical cylinder lead anode placed upstream from the cathode and flush with it. The cathodic mass transfer coefficient was increased by a factor of 2.75–6.7 over the natural convection value depending on the rate of oxygen discharge at the lead anode and height of the cathode. The data were correlated by the equation:J=0.66(F_rR_e)^–0.21 An electrochemical reactor built of a series of vertical coaxial annular cells stirred by the counter electrode gases is proposed as offering an efficient way of stirring with no external stirring power consumption.Nomenclature a, b, c constants - C concentration of copper sulphate, mol cm^–3 - d cylinder diameter, cm - D diffusivity, cm² s^–1 - F Faraday's constant - g acceleration due to gravity, cm² s^–1 - h electrode height, cm - i current density at the oxygen generating anode, A cm^–2 - I _L limiting current density, A cm^–2 - K mass transfer coefficient, cm s^–1 - P gas pressure, atm - R gas constant, atm cm³ mol^–1 K^–1 - T temperature, K - u solution viscosity, poise - V oxygen discharge rate as defined by Equation 9, cm³ cm^–2 s^–1 or cm s^–1 - Z number of electrons involved in the reaction - J mass transferJ factor (S _tS_c ^0.66) - S _t Stanton number (K/V) - S _c Schmidt number (v/D) - S _h Sherwood number (Kh/D) - R _e Reynold's number (/Vh/u) - F _r Froude number (V ² /hg) - G_r Grashof number [gh ³/v² (1–)] - density of the solution, g cm^–3 - kinematic viscosity, cm² s^–1 - void fraction of the gas in the liquid-gas dispersion     

Mass transfer between rough surfaces and solutions containing drag-reducing polymers

Rates of electrochemical mass transfer were measured between finned rotating cylinders and solutions containing drag-reducing polymers. Variables studied were: Reynolds number, polymer concentration and fin height. Polyox and carboxymethyl cellulose (CMC) were used as drag-reducing polymers with concentrations ranging from 10–100 ppm for polyox and from 10–500 ppm for CMC. Cylinders with longitudinal fins ofe/d ranging from 0·0185–0·075 were used. Reynolds number was varied between 1000–10000. It was found that the presence of fins on the cylinder surface reduces the adverse effect of the polymer on the rate of mass transfer, the higher the fin height the lower is the ability of the polymer to reduce the rate of mass transfer. Mass transfer data for solutions containing polyox were correlated by the equation: (St) = 0.765(Re)^-0.36(Sc)^–0.669(e/d)^0.36 Mass transfer data for solutions containing CMC were correlated by the equation: (St) = 1.704(Re)^–0.36(Sc)^–0.75(e/d)^0.315 List of symbols I _L limiting current density based on the projected area of the electrode (A cm^–2) - K mass transfer coefficient (cm s^–1) - Z number of electrons involved in the electrode reaction - C ferricyanide concentration (mol cm^–3) - F Faraday's constant - u dynamic viscosity (g cm^–1 s^–1) - solution density (g cm^–3) - angular velocity (rad s^–1) - V peripheral velocity (cm s^–1) - D diffusion coefficient of ferricyanide ion (cm² s^–1) - d cylinder diameter (cm) - e fin height (cm) - (Sc) u/(D), Schmidt number - (Re) vd/u, Reynolds number - (St) K/V, Stanton number