Calculation of limiting current density of metal electrodeposition on vertical plane electrode under conditions of natural convection

An approximate analytical solution of problem of ion transfer near a vertical plane electrode surface is obtained for the metal electrodeposition proceeding at the limiting current from electrolyte containing ions of three types under the conditions of natural convection. In contrast to previous studies, no transport numbers are used here, and the migration transfer of electroactive electrolyte component is taken into account. The equations obtained take into consideration the effect of supporting electrolyte concentration and its migration on the limiting current. The limiting current densities (mass-transfer coefficients), which are calculated by equations proposed here and by the finite-difference method, are compared.     

The vertical distribution of current in a gas-evolving membrane cell

A one-dimensional numerical model to describe gas void fraction and current distribution in five model membrane cell configurations is described in this work. The five models describe ideal (equipotential), upright (top cathode/bottom anode), inverted, u- and n-type electrical connections with anodic chlorine and cathodic hydrogen evolution in each case. In all but the first case the finite resistances of the electrodes are taken into account. The effects of (a) different terminal arrangements, (b) different current densities, (c) different cell heights, (d) different compartment widths, and (e) different overvoltages, have been investigated. For each study the current distribution and anolyte and catholyte void fraction distribution is displayed. The resistive components of the cell voltages are also calculated; the calculated resistive voltage loss varies between extremes of 0.291 V for the ideal cell to 0.377 V for the inverted cell at 3 kA m^–2 and 0.25 m cell height with typical fixed values of other parameters.Nomenclature A cross-sectional area - d bubble diameter - void fraction - _m maximum void fraction - G gas volumetric flow rate - K ratio of conductivities of bubble-free and bubble-filled electrolyte - L liquid volumetric flow rate - electrolyte viscosity - R _AN,R _A resistances of anode, anolyte, membrane - R _M,R _C cathode and catholyte, respectively (see - R _CA resistive network scheme of Fig. 2) - _L, _G liquid and gas phase densities - u ₁ single bubble rise velocity - u _sw bubble swarm rise velocity 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.     

Mass transfer under combined natural and forced convection in a vertical flow channel: Assisting and opposing flows

Average and local mass-transfer coefficients were measured by the limiting current technique during zinc electrodeposition from aqueous ZnSO₄ solution under combined laminar natural and forced convection in a vertical flow channel. For assisting flow, the results agree with the theoretical correlation Sh³ = SH_F³ + SH_N³. For opposing flow, the experimental data indicate flow reversal and support the theoretical assumptions of superimposition of velocity gradients and the distinction between the forced and natural-dominant regions, below and above the point of zero shear stress.     

The effect of the gas void distribution on the ohmic resistance during water electrolytes

Due to the presence of gas bubbles on the electrode surface and in the interelectrode gap during water electrolysis, the ohmic resistance in the cell increases. The main aim of this investigation is to obtain insight into the effect of the gas void distribution on the ohmic resistance in the electrolysis cell. The gas void distribution perpendicular to the electrode surface has been determined at various current densities, solution flow velocities and heights in the cell, taking high speed motion pictures. From these measurements it follows that two bubble layers can be distinguished. The current density distribution and the ohmic resistance in the electrolysis cell have been determined using a segmented nickel electrode. The current density decreases at increasing height in the cell. The effect is more pronounced at low solution flow velocities and high current densities. A new model to calculate the ohmic resistance in the cell is proposed.Nomenclature A _l electrolyte area (m²) - c constant (–) - d _wm distance between the working electrode and the diaphragm resp. the tip of the Luggin capillary (m) - E voltage of an operating cell (V) - f gas void fraction (–) - F Faraday constant (C/mol) - f ₀ gas void fraction at the electrode surface (–) - f _b gas void fraction in the bulk electrolyte (–) - h height from the bottom of the working electrode (m) - h _r reference height (= 1 cm) (m) - H total height of the electrode (m) - i current density (A m^–2) - i _av average current density (A m^–2) - i _r reference current density (= 1 kA m^–2) (A m^–2) - R resistance () - R specific resistance (_m) - R unit surface resistance (m²) - R ₁ resistance of the first bubble layer () - R ₂ resistance of the second bubble layer () - R _cell ohmic resistance in the cell () - R _b bubble radius (m) - s _l degree of screening by bubbles in the electrolyte (–) - _l liquid flow velocity (m s^–1) - _{1, r} reference liquid flow velocity (= l m s^–1) (m s^–1) - V _M molar gas volume (m³ mol^–1) - w width of the electrode (m) - x distance from the electrode surface (m) - thickness of the bubble layer adjacent to the electrode (m) - number of bubbles generated per unit surface area and unit time (m^–2 s^–1) Paper presented at the International Meeting on Electrolytic Bubbles organised by the Electrochemical Technology Group of the Society of Chemical Industry, and held at Imperial College, London, 13–14 September 1984.     

Mass or heat transfer coefficients under assisting or non-assisting natural and forced convection near a vertical wall

The velocity superimposition and the Lighthill transformation are used to calculate heat or mass transfer coefficients for large Schmidt (Prandtl) number under combined natural and forced convection near a vertical wall at constant concentration or temperature. The resultant correlation for assisting flow is in complete agreement with the results of Ruckenstein and Rajagopalan and Churchill. However, the results for non-assisting (opposing) flow are in disagreement with the simplified result of Ruckenstein and Rajagopalan due to the occurrence of zero shear stress and flow reversal at the wall. An explicit, but more complicated, correlation in terms of the pure natural and pure forced convection is presented for the case of opposing flow.     

Effect of ambient conditions on performance and current distribution of a polymer electrolyte membrane fuel cell

The performance and current distribution of a free-breathing polymer electrolyte membrane fuel cell (PEMFC) was studied experimentally in a climate chamber, in which temperature and relative humidity were controlled. The performance was studied by simulating ambient conditions in the temperature range 10 to 40 °C. The current distribution was measured with a segmented current collector. The results indicated that the operating conditions have a significant effect on the performance of the fuel cell. It was observed that a temperature gradient between the fuel cell and air is needed to achieve efficient oxygen transport to the electrode. Furthermore, varying the air humidity resulted in major changes in the mass diffusion overpotential at higher temperatures.     

Electrochemical characterization of tenoxicam using a bare carbon paste electrode under stagnant and forced convection conditions

From potentiostatic current transients and voltammetry studies, carried out under both stagnant and forced convection conditions, the tenoxicam electrochemical behavior on a bare carbon paste rotating disk electrode was assessed in an aqueous solution (pH = 0.403). It was found that tenoxicam's electrochemical oxidation is a mass transfer-controlled process where a current peak is clearly formed at around 0.74 V when the potential scan was varied in the positive direction. However, when the potential was switched to the negative direction, up to the initial potential value, no reduction peak was formed. Tenoxicam's electrochemical oxidation follows an EC mechanism where the electrodic and chemical kinetics are fast. From sample-current voltammetry both the number of electrons, n, that tenoxicam losses during its electro-oxidation and its half-wave potential, E_1/2, were determined to be 2 and 0.770 V vs. Ag/AgCl, respectively. Moreover, from differential pulse voltammetry plots it was confirmed that effectively in this case n = 2. Considering 2 electrons and both the Randles-Sevcik and Cotrell equations, the tenoxicam's diffusion coefficient, D, was determined to be (3.745 ± 0.077) × 10^?6 and (4.116 ± 0.086) × 10^?6 cm² s^?1, respectively. From linear sweep voltammetry plots recorded under forced convection conditions, it was found that Levich's equation describes adequately the limiting current recorded as a function of the electrode rotation rate, from where the D value was also found to be (4.396 ± 0.058) × 10^?6 cm² s^?1. Therefore, the average D value was (4.09 ± 0.33) × 10^?6 cm² s^?1. Furthermore, from the radius of the tenoxicam molecule, previously optimized at M052X/6-31 + G(d,p) level of theory, and using the Stokes–Einstein approach, D was also estimated to be 4.54 × 10^?6 cm² s^?1 which is similar to the experimentally estimated values, under both stagnant and forced convection hydrodynamic conditions.     

A hydrodynamic model for the ohmic interelectrode resistance of cells with vertical gas evolving electrodes

A hydrodynamic model of a cell with gas evolving vertical electrodes is developed. The leading assumption is a separation of the ohmic resistance of the dispersion of gas bubbles in electrolyte between the electrodes into two parts: a stagnant boundary layer at the electrode(s) being enriched in gas and a flowing bulk in the centre region. The mathematical treatment of the model leads to equation (18) which allows to predict the mean ohmic resistance of the dispersion in the cell. Results are compared with experimental data.     

Variations of cell resistance and current distribution with current feeder configurations

Two models of current feeder configurations for resistive electrodes are presented, one in which the feeders were connected at the same end of a pair of electrodes of a unit cell and one in which connection was made at opposite ends. Expressions for the cell resistance and the current distribution were derived for these two current feeder configurations on the assumption of a linear type of overpotential. The cell resistance in the ‘opposite ends’ configuration was larger than that for the ‘same ends’ arrangement. Conversely, the current distribution in the former was more uniform than that in the latter. The relation between the total cell resistance and the number of current feeders,n, was obtained. An increase inn led to a decrease in the resistive loss of the electrodes by an amount corresponding to 1/n ², irrespective of current feeder configuration, when the resistance of the electrode was not so great as that of the solution.     

Behaviour of a tall vertical gas-evolving cell. Part I: Distribution of void fraction and of ohmic resistance

Electrolysis of a 22 wt % NaOH solution has been carried out in a vertical tall rectangular cell with two segmented electrodes. The ohmic resistance of the solution between a segment pair has been determined as a function of a number of parameters, such as, current density and volumetric rate of liquid flow. It has been found that the ohmic resistance of the solution during the electrolysis increases almost linearly with increasing height in the cell. Moreover, a relation has been presented describing the voidage in the solution as a function of the distance from the electrodes and the height in the cell.Notation A _e electrode surface area (m²) - a _s parameter in Equation 12 (A^–1) - b _s parameter in Equation 12 - d distance (m) - d _ac distance between the anode and the cathode (m) - d _wm distance between the working electrode and an imaginary separator (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) - h _s height of a segment of working electrode (m) - I current (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 _x-19 total current for the segment pairs fromx to 19 inclusive (A) - i current density A m^–2 - N _s total number of gas-evolving pairs - n ₁ constant parameter in Equation 8 - n _a number of electrons involved in the anodic reaction - n _c number of electrons involved in the cathodic reaction - 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 - S _b bubble-slip ratio - S _b,20 S _b at segment pair 20 - S _b,h S _b at heighh in the cell - T temperature (K) - V _m volume of 1 mol gas saturated with water vapor (m³ mol^–1) - 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) - W _e width of electrode (m) - X distance from the electrode surface (m) - Z impedance () - Z real part of impedance () - Z imaginary part of impedance () - resistivity of solution ( m) - _p resistivity of bubble-free solution ( m) - gas volumetric flow ratio - ₂₀ at segment pair 20 - _s specific surface resistivity ( m²) - _{s, p s} for bubble-free solution ( m²) - thickness of Nernst bubble layer (m) - ₀ ath=0 (m) - voidage - _x,0 atx andh=0 - _0,0 voidage at the leading edge of electrode wherex=0 andh=0 - _,h voidage in bulk of solution at heighth - ₂₀ voidage in bubble of solution at the leading edge of segment pair 20     

Electrochemical mass transfer at a fixed bed of spheres under forced convection induced by counterelectrode gas bubbles

Rates of mass transfer were measured at a cathode consisting of a fixed single layer of spheres stirred by oxygen evolved at a horizontal disc anode placed below. The rate of mass transfer was found to increase by a factor of 2–6.5 over the natural convection value, depending on the operating conditions. A mathematical model based on the surface renewal theory was formulated to explain the mechanism of mass transfer at gas stirred electrodes. A new electrochemical reactor built of a series of cells arranged vertically in a cylindrical container, each cell consisting of a screen gas-evolving counterelectrode placed below a fixed bed working electrode, is proposed as offering an efficient way of stirring with no external stirring power consumption.     

Modelling and calculation of the current density distribution evolution at vertical gas-evolving electrodes

During industrial electrolysis, for hydrogen, dichloride or aluminium production, there is bubbles creation at one or two electrodes which imply a great hydrodynamic acceleration but also a quite important electrical field disturbance. This disturbance can lead to the modification of the local current density and to anode effects for example. There is few works concerning the local modelling of coupled electro active species transport and electrochemical processes in a biphasic electrolyte. There are also few local experimental measurements in term of chemical composition, temperature or current density which would allow the numerical calculations validation. Nevertheless, effects like the anode effect, particularly expensive on the point of the process efficiency, should need a better understanding. Nowadays, the respective roles of the local temperature increases, the electro active specie composition or the transport properties modification due to bubbles are not known.The goal of the present work is the modelling and the numerical simulation of the vertical electrode configuration for a biphasic electrolysis process. Bubbles presence is supposed to modify the electrical properties, and then the electro active species diffusive transport and the current density. Bubbles are also motion sources for the electrolysis cell flow, and then hydrodynamic properties are strongly coupled with species transport and electrical field. The present work shows hydrodynamic and electrical properties in a laboratory scale electrolysis cell with a vertical electrode. The numerical algorithm used was the finite volume used in the computational fluid dynamic software Fluent^®.     

Mathematical modeling of radial impurity distribution in crystals grown under laminar convection conditions

A mathematical model of the process of the growth of semiconductor crystals by the Bridgman method is developed. In modeling of a known space experiment, the character of convective flows and their influence on the of the axial and radial impurity distribution during the growth of a crystal are studied. As a result of numerical calculations, the possibility of formation of the large radial nonuniformity of the impurity distribution observed in a number of space experiments is shown.     

Potential distribution in flow-through porous electrodes under limiting current conditions

This paper presents an experimental study of metal— solution potential distributions in flow- through porous electrodes of fixed beds of spherical conducting particles.The distributions are determined when the electrodes are working at the limiting diffusion current. They are shown to be in agreement with the corresponding theoretical distributions.     

Calculations on the effect of gas evolution on the current-overpotential relation and current distribution in electrolytic cells

Gas evolution during electrode reactions has several effects on the electrode behaviour. One of these effects is the nonuniform increase of the resistivity of the electrolyte with the resultant increase of IR drop through the solution and the distortion of current distribution. Calculations of these effects are presented for an electrode built of vertical blades. This geometry has the peculiarity that it allows the inclusion of linear polarization and gas effects in the treatment, without the necessity to use numerical or approximate solutions of the differential equations. It is shown that the system parameters can be combined into a single dimensionless parameter to describe those aspects of the electrode behaviour which depend on the gas evolution. The parameters examined include the geometry of the electrode, the polarization resistance, gas bubble rise velocity, and solution resistivity. Expressions are given for optimization of the electrode geometry to achieve minimum overpotential.Nomenclature b Polarization resistance ( cm²) - C Constant, =RT( + t)/lPtFs (A^–1cm) - E(x) Potential of the solution at pointx (V) - f _av Average volume fraction of gas (dimensionless) - (fy) Volume fraction of gas at heighty (dimensionless) - f(Y) Volume fraction of gas at reduced heightY (dimensionless) - F Faraday number (coulomb mol^–1) - h Height of the electrode (cm) - i Nominal current density of the electrode =I _T/hw (A cm^–2) - i(y) Local electrode current density at heighty (A cm^–2) - i(Y) Local electrode current density at reduced heightY (A cm^–2) - i _f(x) Faradaic current density at pointx (A cm^–2) - i _f(X) Faradaic current density at reduced lengthX (A cm^–2) - i _f,av Average faradaic current density in the slot=I _s/2hl(Acm^–2) - I _s Total current entering one slot (A) - I _T Total current flowing to the electrode (A) - I(x) Current flowing in the solution phase of one slot at pointx (A) - k Constant, = (2/b)^1/2 (cm^–1) - K Dimensionless parameter =hRT(2/b)^1/2/4lPzFs, or = 1–(1–iCh)^1/4 - l Horizontal length of the slot (cm) - n Number of slots on the electrode (dimensionless) - p Pressure of gas liberated on the electrode (assumed to be independent of height) (atm) - R Universal gas constant (cm³ atm K^–1 mol^–1) - s Bubble rise velocity (cm s^–1) - t Thickness of the blades (cm) - T Temperature of the gas (K) - dV(y) Volume of gas present in a volume element of the slot (cm³) - w Width of the electrode (cm) - x Horizontal distance from the back plate (cm) - X Reduced horizontal distance =x/l (dimensionless) - y Vertical distance from the bottom of the electrode (cm) - Y Reduced vertical distance =y/h (dimensionless) - z Number of Faradays needed to produce one mole of gas (mol^–1) - Width of a slot (blade spacing) (cm) - Measured overpotential of the electrode =(l)(V) - (x) Overpotential at pointx (V) - Resistivity of gas free electrolyte ( cm) - (y) Resistivity of gas filled electrolyte at, heighty ( cm).     

Mass transfer and shear stress on a vertical electrode with gas evolution

The relationship between the mass transfer coefficient and the shear stress along the vertical electrode was investigated under electrolytic gas evolution, either oxygen or hydrogen, from alkaline solution containing ferrocyanide and ferricyanide ions. The shear rate obtained from the shear stress measurement was empirically correlated with the gas evolving current density, the electrode height and the liquid kinematic viscosity. The dimensionless correlation of the mass transfer coefficient with the shear stress under oxygen evolution condition agreed formally with the correlation of the mass transfer in turbulent-free convection. On the other hand, the experimental results under hydrogen evolution varied greatly from those under oxygen evolution.     

Effect of contact resistance between particles on the current distribution in a packed bed electrode

A microscopic, modelistic approach was carried out to elucidate the electrochemical reduction of a nuclear waste solution in a packed bed electrode. The interfacial surface reactions within the packed bed were taken into account and the particle–particle contact resistance through oxide films was found to be big enough to effect the potential distribution throughout the bed. On the basis of equations developed here, the contribution of the resistance of the oxide film to the potential and current distribution throughout the bed was compared with the macroscopic homogeneous approach.     

Mass transfer study of the electropolishing of vertical cylinders with active ends under natural convection conditions

Rates of electropolishing of vertical copper cylinders with active ends in H₃PO₄ were studied by measuring the limiting current under natural convection. Variables studied were H₃PO₄ concentration, cylinder diameter and aspect ratio. The rate of polishing of the whole cylinder was represented by the mass transfer equation $$Sh = 0.33(Sc Gr)^{0.32}$$ for the range 1.17 × 10¹⁰ < Sc Gr < 5.11 × 10¹¹. Rates of mass transfer were measured also at vertical cylinder with insulated ends, and the upward facing surface (disc). Data for the vertical cylindrical surface were represented for the range 8.75 × 10⁹ < Sc Gr < 1.1 × 10¹² by the equation $$Sh = 1.206(Sc Gr)^{0.255}$$ while data at the upward facing disc were correlated for the range 0.11 × 10¹⁰ < Sc Gr < 46 × 10¹⁰ by the equation $$Sh = 0.17Sc^{0.396} (Sc Gr)^{0.146}$$ A comparison between the measured rate of mass transfer at the whole cylinder and the value calculated by adding the rates of mass transfer at the separate surfaces of the cylinder shows that the measured value deviates from the calculated value, the degree of deviation increases with increasing Sc × Gr. Deviation was attributed to flow interaction at the different cylinder surfaces.     

Ohmic resistance of solution in a vertical gas-evolving cell

Vertical electrolysers with a narrow cell gap between a gas-evolving electrode and a membrane or diaphragm are used to produce industrial gases. Generally, the local current density decreases with height in the cell. Electrolyses are carried out with a KOH solution in a tall vertical divided rectangular cell with two gas-evolving electrodes. Either the hydrogen or the oxygen bubbles containing solution from the divided cell are passed through a small measuring cell. Ohmic resistance experiments are carried out in the small measuring cell with a gas-evolving electrode and a gas diffusion electrode, on which no gas bubbles are evolved. The effect of various parameters, viz. current density, solution flow rate and temperature, on the ohmic resistance of solution in the measuring cell are determined. It is found that the normalized ohmic resistance of the solution in the measuring cell during electrolysis increases with current density and with the gas voidage in the bulk of solution, decreases with increasing solution flow rate and is practically independent of temperature at 25 to 60 OC. Moreover, it is found that for an oxygen evolving electrode in a solution containing only oxygen bubbles, as well as for a hydrogen evolving electrode in a solution containing only hydrogen bubbles, the normalized resistance of the solution between the gas-evolving electrode and the nongas evolving electrode is given by a relatively simple empirical relation. A relation is derived describing the gas voidage in the solution as a function of the distance from the gas-evolving electrode in the presence and the absence of gas bubbles in the bulk solution.