Mathematical modelling of electrode growth

It is known that during electrodeposition or dissolution electrode shape change depends on the local current density (Faraday's law in differential form). Assuming that concentration gradients in the bulk of the solution may be neglected, the current distribution in an electrochemical system can be modelled by a Laplace equation (describing charge transport) with nonlinear boundary conditions caused by activation and concentration overpotentials on the electrodes. To solve this numerical problem, an Euler scheme is used for the integration of Faraday's law with respect to time and the field equation is discretized using the boundary element method (BEM). In this way, and by means of a specially developed electrode growth algorithm, it is possible to simulate electrodeposition or electrode dissolution. In particular, attention is paid to electrode variation in the vicinity of singularities. It is pointed out that the angle of incidence between an electrode and an adjacent insulator becomes right (/2). This is confirmed by several experiments.List of symbols x _i coordinates of a point i belonging to a boundary (m) - t time (s) - h thickness variation at a point belonging to an electrode (m) - M molecular weight (kgmol^–1) - m specific weight (kgm^–3) - z charge of an ion (C) - F Faraday's constant (C mol^–1) - R _a2 impedance of the linearized activation overvoltage on cathode (S2 cm^–2) - efficiency of the reaction - electric conductivity (^–1 m^–1) - U electric potential (V) - rate of mechanical displacement of a point (m s^–1) - V applied potential on an electrode (V) - W Wagner number defined as the ratio of the mean impedance of the reaction and the mean ohmic resistance of the cell given by L/ with L a characteristic length of the cell. - overvoltage (V) - ₁ overvoltage on anode (V) - ₂ overvoltage on cathode (V)     

Zirconium tetra-n-butylate modified with different organic acids: Hydrolysis and polymerization of the products

In this work; (a) complexation reaction of zirconium tetra-n-butylate, Zr(OBu ⁿ )₄, with MAc and different organic acids. (b) the hydrolysis reaction of modified Zr species, and (c) the polymerization reaction of complex products are studied. Zr(OBu ⁿ )₄ was reacted with different mole ratios of methacrylic acid (MAc) at room temperature and the maximum combination ratio was found to be 12 [Zr(OBu ⁿ )₄MAc] by FT IR. The modification of zirconium tetra-n-butylate with the acid mixtures [methacrylic acid-acetic acid (MeCOOH), methacrylic acid-propionic acid (EtCOOH), methacrylic acidbutyric acid (PrCOOH)] was made for a combination ratio of 111 [MAcRCOOHZr(OBu ⁿ )₄RMe. Et, Pr] and the products were characterized by¹H-NMR, FT-IR, and UV-spectroscopies. Following their synthesis, hydrolysis of the complexes with various amounts of water and polymerization with benzoyl peroxide were realized. The hydrolysis and polymerization products of the complexes were studied by Karl-Fischer Coulometric titration and thermal analysis respectively. Methyl-ethyl-ketone (MEK) and chloroform were chosen as solvents.     

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 and momentum transfer enhancement due to electrogenerated gas bubbles

The effect of electrogenerated gas bubbles with simultaneous bulk liquid flow on the mass and momentum transfer at a wall of an electrolytic cell is experimentally determined. The local mass transfer coefficient and electrolyte shear stress are obtained using two types of microelectrodes imbedded in the channel wall. The influence of the most important parameters (electrolyte velocity, position along the wall, gas electrogeneration rate) on the transfer enhancement is studied and an analogy between mass and momentum transfer in the presence of bubbles is clearly demonstrated from the experimental results. The comparison with classical correlations, valid for systems involving natural turbulence, shows the higher energetic efficiency of devices where the turbulence is artificially generated by electrolytic gas bubbles.Nomenclature A constant parameter in Equation 3 - ¯C time averaged value of the concentration of a reacting species - c ₀ molar concentration in the bulk of the solution - d microelectrode diameter - d _e hydraulic equivalent diameter - D molecular diffusion coefficient - D _t turbulent diffusivity of mass transfer - f/2 friction factor, =/gr¯v ² - h channel thickness - I _g electrogeneration rate - i _g electrogeneration current density - i _l limiting current density on a microelectrode imbedded in the conducting wall - i_l limiting current density on a microelectrode imbedded in the inert wall - k _d local mass transfer coefficient - k local mass transfer coefficient on a microelectrode in the non-conducting wall - N _M specific mass flux near an interface - Re Reynolds number, = (¯vd _e)/v - s velocity gradient, = (¯v _x/y)y = o - s ⁺ dimensionless velocity gradient, =sd ²/D - Sc Schmidt number, =v/D - Sh Sherwood number, = (k _d x)/D - St Stanton number, =k _d/¯v - ¯v, ¯v _x electrolyte velocity - v ^* friction velocity, = (/)^1/2 - v ⁺ normalized velocity, =¯v _x /v ^* - x axial coordinate - y coordinate perpendicular to the wall - y ⁺ dimensionless length = (yv ^*)/v Greek letters parameter defined in Equation 8 - boundary layer thickness - ⁺ dimensionless form of , = (s/v)^1/2 - , _x electrolyte shear stress - dynamic viscosity - kinematic viscosity - _t momentum transfer diffusivity - specific gravity - ² variance of the fluctuations ofi _L ori _L Paper presented at the International Meeting on Electrolytic Bubbles organized by the Electrochemical Technology Group of the Society of Chemical Industry, and held at Imperial College, London, 13–14 September 1984.     

Some theoretical considerations on the design of a practical anode in an electrochemical reactor

A theoretical analysis of the membrane current distribution is carried out for a typical three-compartment electrolyser in order to point out the effects of geometry on the design of mesh anodes. The factors considered here include the introduction of an insulated border, the perforation of the anode, the finite conductivity of the substrate, and the introduction of a bus bar connection between the anode and the current lead. It is recommended that no insulated border be introduced, since, while reducing the anode area and consequently its cost, it leads to a nonuniform membrane current distribution and hence decreases membrane efficiency. Also, titanium is found to be a suitable substrate for the anode in spite of its relatively low conductivity.Nomenclature a Dummy variable in Equation 3 - b Border width - b ^* Effective border width - f Fraction of open area in electrode - F _B Parameter defined by Equation 4 - F _p Parameter defined by Equation 8 - F _be Parameter defined by Equation 15 - I Total cell current - i Local current density on the membrane at a point - i Current density along the membrane far from the border - _loc Average value of current density over a small portion of the membrane - _cell Average value of current density over the whole membrane - Average value of current density on membrane far from the border - i _max Maximum value of current density on membrane - _loc,max Maximum value of _loc on membrane due to electrode and bus bar resistance effects - i _p Maximum value of current density over a single electrode perforation - j (–1)^1/2 - l _p Characteristic length of mesh - L Dimension of anode in the direction of bus bar orientation - L Dimension of anode in the direction perpendicular to bus bar - L Width of bus bar - s Interelectrode gap - s ₁ Membrane to anode gap - R Electrolyte and membrane resistance - x _b Coordinate along length of bus bar - x _B Coordinate in border effect analysis - x _e Coordinate along electrode in the analysis of its resistance effect - x _P Coordinate in perforation effect analysis - _b Bus bar thickness - _e Electrode thickness - _b Bus bar resistivity - _e Electrode resistivity - _em Resistivity of metal in electrode - _b Potential at a point on the bus bar - _e Potential at a point on the electrode - ¯ _e Average potential over the electrode - _max Potential at the current source - _cath Potential at the equipotential cathode     

Depolarization of the hydrogen evolution reaction on cadmium electrodes in alkaline media

Depolarization of the hydrogen evolution reaction on high purity polycrystalline cadmium electrodes in alkaline media (12pH14 and 5T55°C) produced by cathodization in the range of potential comprised between the Cd/Cd(OH)₂ electrode potential and the net HER potential under solution stirring conditions has been studied. The depolarization effect depends on the perturbing potential programme and it is little affected by the alkaline cation in solution. Results are discussed in terms of three concurrent reactions, namely the electrochemical formation of Cd(OH)₂ and soluble Cd(OH) ₃ ^– , the HER and the electrocrystallization of cadmium renewing the fresh active sites for the HER, SEM micrographs of activated cadmium electrodes reveal a heterogeneous surface topography of the new cadmium layers.     

Flow-through and flow-by porous electrodes of nickel foam Part IV: experimental electrode potential distributions in the flow-through and in the flow-by configurations

Experimental distributions of the solution potential in flow-through and flow-by porous electrodes of nickel foam operating in limiting current conditions are presented. These are in good agreement with the corresponding theoretical distributions. In the case of a flow-by configuration used in a two-compartment cell, the experiments confirm the validity of the models, presented in Part III, which take into account the presence of a separator (ceramic porous diaphragm or ion exchange membrane).Nomenclature a _e specific surface area per unit volume of electrode - C ₀ entrance ferricyanide concentration (y=0) - D molecular diffusion coefficient of ferricyanide - E _e cathode potential - F Faraday number - mean (and local) mass transfer coefficient - L electrode thickness - L _s-L separator thickness - m number of sheets of foam in a stack - n number of terms in Fourier series - Q volumetric flow-rate - r _s ohmic specific resistance of the separator - mean flow velocity based on empty channel - V constant potential - X conversion - x coordinate for the electrode thickness - y coordinate for the electrode length - y ₀ length of the porous electrode - z number of electrons in the electrochemical reaction Greek symbols parameter - parameter - ionic electrolyte conductivity - _sc solution potential in the pores of the cathode - _M matrix potential ( _sc = constant) - parameter [=n/y ₀] - electrolyte density - mean porosity - kinematic viscosity - E _c potential drop in the porous cathode - potential drop defined in Fig. 5 Indices c cathodic - o electrolyte alone - s separator     

Calculation of mean current densities in a two rotating disc electrodes system

A theoretical relationship for mass transfer in the laminar flow region of streaming in a rotating electrolyser was derived by the method of similarity of the diffusion layer for electrodes placed sufficiently far from the rotation axis. The obtained relationship was compared with the known equations valid for systems with axial symmetry. The mean current densities were found from the numerical solution of the convective diffusion equation by the finite-element method and were compared with experimental results.Nomenclature a constant, exponent - c concentration - c ₀ concentration in the bulk phase - C _ij matrix coefficient - D diffusion coefficient - F Faraday constant, 96487 C mol^–1 - h interelectrode distance - j current density - mean current density - J mass flux density - L _j base function - n number of transferred electrons in electrode reaction - n _r outer normal to the boundary - mass flux - N number of nodal points in an element - Q volume rate of flow - mean volume rate of flow - r radial coordinate - r ₀ inner electrode radius - r _l outer electrode radius - r _v radius of inlet orifice - r _d outer disc radius - v _r radial velocity component - v _z normal velocity component - z normal coordinate - thickness of the layer in which the equation of convective diffusion is solved - boundary of the integration domain - thickness of the diffusion layer - _N thickness of the Nernst diffusion layer - v kinematic viscosity - angular velocity - surface Criteria Re _chan channel Reynolds numberQ/hv - Re _loc local Reynolds number,Q/(r + r ₀) - local Reynolds number at mean electrode radius,Q/v(r ₁ +r ₀) - Re _rot rotation Reynolds number, r _d ² /v - modified rotation Reynolds number at mean electrode radius, (r ₁+r ₀)²/4v - Ré _rot modified rotation Reynolds number, (r+r ₀)²/4v - Sc Schmidt number,v/D - Sh _r local Sherwood number,j(r-r ₀)/nFDc _o - mean Sherwood number, - Ta Taylor number,h(/v)^1/2     

Experimental investigation of the primary and secondary current distribution in a rotating cylinder Hull cell

A rotating cylinder cell having a nonuniform current distribution similar to the traditional Hull cell is presented. The rotating cylinder Hull (RCH) cell consists of an inner cylinder electrode coaxial with a stationary outer insulating tube. Due to its well-defined, uniform mass-transfer distribution, whose magnitude can be easily varied, this cell can be used to study processes involving current distribution and mass-transfer effects simultaneously. Primary and secondary current distributions along the rotating electrode have been calculated and experimentally verified by depositing copper.List of symbols c distance between the cathode and the insulating tube (cm) - F Faraday's constant (96 484.6 C mol^–1) - h cathode length (cm) - i local current density (A cm^–2) - i _L limiting current density (A cm^–2) - i _ave average current density along the cathode (A cm^–2) - i ₀ exchange current density (A cm^–2) - I total current (A) - M atomic weight of copper (63.54 g mol^–1) - n valence - r _p polarization resistance () - t deposition time (s) - V _c cathode potential (V) - Wa _T Wagner number for a Tafel kinetic approximation - x/h dimensionless distance along the cathode surface - z atomic number Greek symbols _a anodic Tafel constant (V) - _c cathodic Tafel constant (V) - solution potential (V) - overpotential at the cathode surface (V) - density of copper (8.86 g cm^–3) - electrolyte conductivity ( cm^–1) - deposit thickness (cm) - _ave average deposit thickness (cm) - surface normal (cm)     

An impedance method for the characterization of porous carbons

A method is proposed whereby electrode impedance data may be analysed to yield information about the structure and composition of porous electrode materials. The method is more suitable for comparative investigations than as a technique for obtaining absolute values of the total surface area of a porous solid in contact with an electrolyte.List of symbols A Surface area of the electrode (cm²) - A Apparent specific area of the electrode material (cm²/cm³) - C _dl Capacitance per unit area (F cm^–2) - C Capacitance per unit pore length (F cm^–1) - E ₀ Potential at pore orifice (V) - i ₀ Current at pore orifice (Amp) - l Depth of penetration of signal (cm) - l ₀ Length of pore (cm) - R Resistance of electrolyte per unit pore length (cm^–1) - r Pore radius (cm) - Z ₁ Capacitative impedance per unit pore-length ( cm) - Z ₀ Impedance of pore () - = (R/Z ₁)^1/2 Reciprocal penetration depth (cm^–1) - Electrolyte resistivity ( cm) - 2f wheref = frequency (Hz)     

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)     

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)     

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     

Kinetics of copper electrodeposition in citrate electrolytes

The kinetics of copper electrocrystallization in citrate electrolytes (0.5M CuSO₄, 0.01 to 2M sodium citrate) and citrate ammonia electrolytes (up to pH 10.5) were investigated. The addition of citrate strongly inhibits the copper reduction. For citrate concentrations ranging from 0.6 to 0.8 M, the impedance plots exhibit two separate capacitive features. The low frequency loop has a characteristic frequency which depends mainly on the electrode rotation speed. Its size increases with increasing current density or citrate concentration and decreases with increasing electrode rotation speed. A reaction path is proposed to account for the main features of the reduction kinetics (polarization curves, current dependence of the current efficiency and impedance plots) observed in the range 0.5 to 0.8 M citrate concentrations. This involves the reduction of cupric complex species into a compound that can be either included as a whole into the deposit or decomplexed to produce the metal deposit. The resulting excess free complexing ions at the interface would adsorb and inhibit the reduction of complexed species. With a charge transfer reaction occurring in two steps coupled by the soluble Cu(I) intermediate which is able to diffuse into the solution, this model can also account for the low current efficiencies observed in citrate ammonia electrolytes and their dependencies upon the current density and electrode rotation speed.Nomenclature b, b ₁, b ₁ ^* Tafel coefficients (V^–1) - bulk concentration of complexed species (mol cm^–3) - (si*) concentration of intermediate C^* atx=0 (mol cm^–3) - C concentration of (Cu Cit H)^2– atx=0 (mol cm^–3) - C C variation due to E - C concentration of complexing agent (Cit)^3- at the distancex (mol cm^–3) - C _o concentrationC atx=0 (mol cm^–3) - C _o C _o variation due to E - Cv _s bulk concentrationC (mol cm^–3) - (Cit H), (Cu), (Compl) molecular weights (g) - C _dl double layer capacitance (F cm^–2) - D diffusion coefficient of (Cit)^3- (cm²s^–1) - D ₁ diffusion coefficient of C^* (cm²s^–1) - E electrode potential (V) - f ₁ frequency in Equation 25 (s^–1) - F Faraday's constant (96 500 A smol^–1) - i, i ₁, i ₁ ^* current densities (A cm^–2) - i i variation due to E - Im(Z) imaginary part ofZ - j - k ₁, k ₁ ^* , K₁, K ₁ ^* , K₂, K rate constants (cms^–1) - K rate constant (s^–1) - K ₃ rate constant (cm³ A^–1s^–1) - R _t transfer resistance (cm²) - R _p polarization resistance (cm²) - Re(Z) real part ofZ - t time (s) - x distance from the electrode (cm) - Z _f faradaic impedance (cm²) - Z electrode impedance (cm²) Greek symbols maximal surface concentration of complexing species (molcm^–2) - thickness of Nernst diffusion layer (cm) - , ₁, ₂ current efficiencies - angular frequency (rads^–1) - electrode rotation speed (revmin^–1) - =K ^–1(s) - _d diffusion time constant (s) - electrode coverage by adsorbed complexing species - (in0) electrode coverage due toC _s - variation due to E     

Modelling of gas-evolving electrolysis cells. III. TheiR drop at gas-evolving electrodes

Based on a potentiostatic interrupter technique theiR drop of the bubble layer in front of gas-evolving electrodes of various shapes has been investigated. At small plane electrodes the dependency ofiR drop on electrode inclination has been studied for hydrogen, oxygen and chlorine evolution. In all systems a slightly up-faced orientation results in a gas bubble layer structure of minimumiR drop. Also for expanded metal electrodes of different shapes theiR drop across the electrode diaphragm gap has been studied. The fractional open cross-section and the inclination angle of the electrode blades have been identified as important parameters with respect to the gas diverting effect. These tendencies have also been confirmed for a pilot cell of 1 m height.Nomenclature b' Tafel slope (V) - c ₀ double layer capacity (F cm^–2) - d thickness (cm) - E electrode potential (V) - F Faraday number (96487 As mol^–1) - i current density (A cm^–2) - R area resistance ( cm²) - R gas constant (8.3144 Ws deg^–1 mol^–1) - T temperature (K) - t time (s) - u _g ⁰ superficial gas velocity (cm s^–1) - u _sw swarm velocity (cm s^–1) - U voltage (V) Greek symbols inclination angle (^o) - symmetry factor (1) - _g gas voidage (1) - _m maximum gas voidage. (1) - overvolgate (V) - electrolyte conductivity (S cm^–1) - _g number of electrons (1) 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.     

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).     

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)     

Studies of three-dimensional electrodes in the FMO1-LC laboratory electrolyser

A FMO1-LC parallel plate, laboratory electrochemical reactor has been modified by the incorporation of stationary, flow-by, three-dimensional electrodes which fill an electrolyte compartment. The performance of several electrode configurations including stacked nets, stacked expanded metal grids and a metal foam (all nickel) is compared by (i) determining the limiting currents for a mass transport controlled reaction, the reduction of ferricyanide in 1 m KOH and (ii) measuring the limiting currents for a kinetically controlled reaction, the oxidation of alcohols in aqueous base. It is shown that the combination of the data may be used to estimate the mass transfer coefficient, _L, and the specific electrode area, A _e, separately. It is also confirmed that the use of three dimensional electrodes leads to an increase in cell current by a factor up to one hundred. Finally, it is also shown that the FM01-LC reactor fitted with a nickel foam anode allows a convenient laboratory conversion of alcohols to carboxylic acids; these reactions are of synthetic interest but their application has previously been restricted by the low rate of conversion at planar nickel anodes.Nomenclature A _e electrode area per unit electrode volume (m²m^–3) - c bulk concentration of reactant (mol m^–3) - E electrode potential vs SCE (V) - E _1/2 half wave potential (V) - F Faraday constant (96 485 C mol^–1) - I current (A) - IL limiting current (A) - j _L limiting current density (A m^–2) - _L mass transfer coefficient (m s^–1) - n number of electrons transferred - p empirical constant in Equation 2 - P pressure drop over reactor (Pa) - R resistance between the tip of the Luggin capillary and the electrode surface () - q velocity exponent in Equation 2 - (interstitial) linear flow rate of electrolyte (ms^–1) - V _e volume of electrode (m³)     

Calorimetric and rheological studies of 12-hydroxystearic acid / digycidyl ether of bisphenol A blends

Summary Above a concentration c^* close to 0.3 wt%, blends of 12-hydroxystearic acid (HSA) with diglycidyl ether of bisphenol A (DGEBA) prepolymer mixed at 80°C give thermally reversible physical gels (organogels) on cooling.According with the literature, the turbidity of the gels indicates fibres of rectangular cross-sectional shapes. The slope of the linear melting heats versus concentration is equal to the melting heat of the pure HSA (-182 ± 4 J.g^-1).The blends are gels as the elastic modulus G is about ten times larger than the loss one G and G is practically independent of the frequency at a given concentration.The sharp variation of the temperature of the endothermic peak T_peak, of the time to reach the rubbery plateau t_r, of the exponent (G) and of the limiting strain _l of the linear viscoelastic domain below 2.5 wt%, is attributed to smaller crystallites in the blend.At a given frequency, G follows a scaling law with the concentration ( ), the scaling exponent ₁ being equal to 3.87 ± 0.02 at 1 rad.s^-1. This indicates that the gel structure is independent of the concentration.     

Impedance spectroscopy study of electrochromism in sputtered iridium oxide films

The a.c. impedance response of sputtered iridium oxide films (SIROFs) was studied at room temperature in 1M H₂SO₄ between 1mHz and 50mHz. The spectra were recorded as a function of applied potential in the range of electrochromic properties from 0.0 to 1.0V vs SCE and before and after an electrochemical treatment consisting of alternatively colouring and bleaching the electrode. The spectra were analysed with help of an equivalent circuit. Between 0.4 and 1.0V, the spectra can be interpreted as due to electrochemical proton insertion in a single phased compound. From the data, hydrogen chemical diffusion coefficients with values ranging from 2 × 10^–8 to 1.1 × 10^–7cm²s^–1 are found. It is shown that this parameter increases fourfold after the cycling treatment and significantly decreases with the amount of inserted hydrogen. Below 0.4V spectrum changes are observed over the intermediate frequency range studied, indicating some changes of the interfacial reactivity which remain to be clarified.