A simple additivity relation for analysis of heterogeneous catalytic reactors with first order reaction

An isothermal heterogeneous reactor with first order kinetics is characterized by two virtual maximum efficiencies, $\text{N}$_kin and $\text{N}$_tr, which are readily determined from the appropriate kinetic and transport parameters. The actual efficiency of the reactor, which is related to the conversion, X, by X = 1?e^?$\text{N}$, can be accurately approximated as $\text{N}$^?1 = $\text{N}$^?1_tr + $\text{N}$^?1_kin. The overall additivity relation offers a simple approach either to predict the conversion or to extract the kinetic or transport parameters when the reactor behavior is affected by both reaction and diffusion.     

Mechanisms and rate laws for oxygen exchange on mixed-conducting oxide surfaces

Transition metal oxides with mobile electrons and oxygen ions (mixed conductors) constitute a broad class of materials that include selective oxidation catalysts, high-temperature electrocatalysts, and ion-transport membrane materials. Although the thermodynamic and transport properties of mixed conductors are generally understood, a consensus has not yet emerged regarding the mechanisms and rate laws governing exchange of oxygen with the bulk at the gas-exposed surface. To aid interpretation of existing kinetic data, and generate testable hypotheses for further research, this paper outlines a framework for predicting O₂ reduction rate laws based on specific reaction mechanisms. Based on nonequilibrium thermodynamics and transition state theory, this framework yields rate laws that are rigorously consistent with thermodynamics, yet allow rates of individual steps to be developed in terms of simple mass action laws, where appropriate. This framework is used to reexamine equilibrium oxygen-exchange kinetics reported for electron-rich perovskite mixed conductors La_1−xSr_xCoO_3−δ (LSC) and La_1−xSr_xFeO_3−δ (LSF), which have a metallic and a semiconducting band structure, respectively. Our analysis suggests that metallic band structure may play an important role in catalysis by stabilizing physisorbed O₂ on the surface. We also show that equilibrium surface exchange rates (as measured by ¹⁸O/¹⁶O isotopes, concentration steps, impedance, etc.) are generally only weak indicators of mechanism, and emphasize the need for kinetic data involving moderate to large displacements from equilibrium.     

Kinetic model of CaO/fly ash sorbent for flue gas desulphurization at moderate temperatures

Characteristics of the sulphation reaction between SO₂ and CaO/fly ash sorbent were analyzed based on TGA results to develop a kinetic model for a dry moderate temperature (400–800 °C) FGD process. It was found that SO₂ diffusion within sorbent particles involved three sub-processes: inter-particle diffusion, inter-grain diffusion and diffusion through product layers and the diffusion dominated the whole sulphation reaction process. The activation energy for product layer diffusion E_diff of 49.3 kJ mol⁻¹ being greater than the chemical reaction activation energy E_a of 13.9 kJ mol⁻¹ verified the importance of the diffusion. Predictions using the kinetic model in which k₀ varies with temperature agree well with the experimental data.     

Reaction kinetics to infer the effect of dopants on ion transport - A case study for Mo+6 doped lithium titanates (Li2TiO3-δ and Li4Ti5O12-δ)

In this paper, a unique approach to correlate influence of doping on ionic mobility, through thermo-kinetic analysis, is reported. Formation kinetics of Li₂TiO₃ and Li₄Ti₅O₁₂, with Mo⁺⁶ doping, were successfully analyzed in ultra-pure Ar atmosphere using differential scanning calorimetry. The results were compared with formation kinetics of pure Li₂TiO₃ and Li₄Ti₅O₁₂ under identical conditions. Field emission scanning electron microscopy (FE-SEM) with electron diffraction spectroscopy (EDS), X-ray diffraction and Raman spectroscopy were employed for the characterization of resulting phases and presence of oxygen vacancy. The results indicate that for doped samples, oxygen vacancy concentration was reduced due to the charge compensation mechanism of the doped ion. The activation energy (E_α) of the different reactions with and without Mo⁺⁶ doping was determined by Kissinger-Akahira-Sunose method. The most probable reaction mechanism was predicted through Master plot approach. The reaction rate controlling step shifted from three-dimensional diffusion (D₃) for undoped Li₂TiO₃ to a chemical reaction (F_n) for doped Li₂TiO₃. For Li₄Ti₅O₁₂ the reaction mechanism (or rate controlling step) was a chemical reaction (F_n) for undoped and nucleation (A_n) for doped material. The results show that diffusion of ions becomes faster in the Mo⁺⁶ doped materials by reducing the charge transfer resistance. Finally, the thermodynamic functions of the transition complex were calculated from kinetic triplets and correlated with thermo-kinetic data.     

On the formulation of the diffusion coefficient in isothermal binary systems

The diffusion of a component in a binary isothermal mixture can be regarded as arising either from its concentration gradient or from the gradient of its chemical potential. The ‘kinetic’ and the ‘thermodynamic’ approach can both be related to the simple, measurable, mutual diffusion coefficient D_m. When considering the concentration dependence of D_m the ‘thermodynamic’ approach leads naturally to the presence of the term (d ln a_i/d ln n_i), while the ‘kinetic’ approach does not. In systems showing marked non-ideality, of which two examples are discussed, allowance for this term greatly simplifies the concentration dependence of D_m. This lends support to the proposition that the gradient of chemical potential should be regarded as the ‘natural force for diffusion’. In ideal systems it does not matter which approach is used, but where non-ideality is marked, the ‘thermodynamic’ approach would appear to have advantage.     

Kinetics,equilibrium and thermodynamics studies of arsenate adsorption from aqueous solutions onto iron hydroxide

This paper reports the equilibrium, kinetics and thermodynamic studies of arsenate adsorption onto freshly precipitated iron hydroxide. Adsorption of arsenate onto iron hydroxide depends on arsenate concentration, contact time and temperature. The intrapartical diffusion model indicates that both the film and intrapartical diffusion control arsenate adsorption on iron hydroxide. The values of activation energies (E_a) indicate the chemical nature of adsorption accompanied by diffusion controlled processes as the rate limiting step. The endothermic nature of the adsorption process suggests that adsorption reaction consumes energy. The negative values of ΔS^# can be assigned to decreased randomness at the solid liquid interface.     

Model‐free kinetics: Curing behavior of phenol formaldehyde resins by differential scanning calorimetry

Isoconversional analysis was used to treat nonisothermal DSC data and yield the dependence of activation energy on conversion during the curing process of PF resins. The shape of the dependence revealed that the curing process of PF resins displayed a change in the reaction mechanism from a kinetic to a diffusion regime. In the kinetic regime a comparative DSC experimental analysis between monomer mixtures and PF resins showed that the addition reactions between phenol and formaldehyde had been mostly completed during the synthesis of PF resins and that the main kinetic reactions contained parallel condensations in the curing process. For the diffusion regime a modified equation for the diffusion rate constant, k_D = D₀ exp(?E_D /RT + K₁α + K₂α²), is proposed. This equation is in good agreement with the experimental dependence of E_α on α in the diffusion regime, which shows the effect of both temperature and conversion on diffusion. A prediction of the conversion advancement with the reaction time under isothermal condition for PF resin has been made. This prediction can be useful in practical applications for evaluating isothermal behavior of thermosetting systems from nonisothermal experimental data. © 2002 Wiley Periodicals, Inc. J Appl Polym Sci 87: 433–440, 2003     

Local current densities in a two rotating disc electrode system with an axial electrolyte inlet measured by an autoradiographic method

The distribution of local current densities in a rotating electrolyser with axial electrolyte inlet in a laminar flow regime was studied by the autoradiographic method. With all systems studied, the local current density decreased monotonically from a maximum value at the inner to a minimum at the outer boundary of the electrode. Experimental results are compared with the numerical solution of the convective diffusion equation by the finite element method.Nomenclature a, b constants - c concentration - c ₀ concentration in the bulk phase - d optical density - D diffusion coefficient - F Faraday constant, 96 487 C mol^–1 - h interelectrode distance - j local current density - J _{n, dif} density of diffusion flux in outer normal direction - n number of electrons transferred in the elementary step - diffusion flux - Q volume rate of flow - r radial coordinate - r ₀ inner electrode radius - r ₁ outer electrode radius - r _v radius of inlet orifice - r _d outer disc radius - t time - v _r radial velocity component of liquid - v _z normal velocity component of liquid - z normal coordinate - thickness of the layer in which the equation of convective diffusion is solved - v kinematic viscocity - angular velocity     

Electrochemical reduction of silver thiosulphate complexes Part II Mechanism and kinetics

In this paper the thermodynamic data for complex formation between Ag⁺ and S₂O_{3 2–} ions, determined previously, are applied to kinetic investigation of the reduction of silver thiosulphate complexes. Both electrochemical (linear sweep voltammetry on a rotating disc electrode) and surface analytical (Auger electron spectroscopy) techniques are used. The deposits resulting from the electrodeposition of silver thiosulphate complexes are shown to be composed of silver and to be polycrystalline. The reduction follows a mechanism involving mass and charge transfer and chemical reaction steps. The relevant kinetic parameters are calculated and a rate equation describing the kinetics of the reduction is given.List of symbols a activity (M) - c concentration (M) - j current density (A m^–2) - j _c current density of charge transfer (A m^–2) - j _m current density of mass transfer (A m^–2) - k rate constant (m s^–1) - y activity coefficient (molarity scale) - D diffusion coefficient against gradient of concentration (m² s^–1) - D diffusion coefficient against gradient of electrochemical potential (m² s^–1) - E electrode potential vs NHE (V) - I ionic strength (M) - T temperature (K) Greek symbols a transfer coefficient - _1n stability constant of Ag(S₂O₃) _n (^2n–1)- - kinematic viscosity (m² s^–1) - rotation speed of the electrode (rad s^–1) Indices b bulk of the solution - f free (= uncomplexed) - 1,n related to complex Ag(S₂O₃)_n ^(n–1) - t total Constants F Faraday constant (96486 A s mol^–1) - R universal gas constant (8.3145 Jmol^–1 K^–1)     

A model for SOFC anode performance

A model for anode performance of a planar anode-supported SOFC is developed. The model includes Butler–Volmer relation for the hydrogen oxidation, Ohm’s law for ionic current and equation of hydrogen mass balance in the anode channel. We show that the regime of anode operation depends on the relation between the cell current density j and the critical current density j_crit. Analytical solutions to the system of governing equations for the case of “low” (j<j_crit) and “high” (jj_crit) currents are derived. In the “low-current” regime the anode polarization voltage is proportional to cell current, which justifies the notion of anodic activation resistivity R_a. Full hydrogen utilization increases the value of R_a by a factor of 2. In the “high-current” regime polarization voltage depends on cell current logarithmically, with the effective Tafel slope being twice the kinetic value (doubling of Tafel slope). In this regime 100% hydrogen utilization leads to a constant 230-mV shift of polarization curve as a whole.     

Recycle reactor models for complex electrochemical/chemical reaction systems

The performance of complex electrochemical reaction sequences in recycle plug flow reactors is mathematically modelled. The reactions include successive electron transfers (EE reactions), chemical reaction interposed between successive electron transfers (ECE reactions), simultaneous electron transfers and simultaneous electron transfer and chemical reaction. Both potentiostatic and galvanostatic operations are considered and the effects of important parameters such as mass transport coefficient, recycle ratio and chemical reaction rate in the recycle loop are highlighted. This is done by considering two important electro-organic synthesis reactions, the production ofp-aminophenol and the reduction of oxalic acid to glyoxylic acid.Nomenclature a activity factor - C _ji concentration of species j at reactor inlet - C _j bulk concentration of species j - C _j ^s surface concentration of species j - C _j ^e concentration of component j returned to reactor inlet stream - C _j concentration from reactor outlet - C _je equilibrium concentration - E electrode potential - F Faraday number - i _n partial current density of stepn - i _T total current density - k deactivation rate constant - k _f n forward electrochemical rate constant of stepn - k _b n backward electrochemical rate constant of stepn - k _f forward chemical reaction rate constant - k _r reverse chemical reaction rate constant - k _Lj mass transfer coefficient for species j - k _a forward adsorption rate constant - k _d reverse adsorption rate constant - L reactor length - r recycle ratio - t reaction time - u velocity - Q flow rate - x reactor dimension - _n constant describing potential dependency of reverse reaction rate constant - _n constant describing potential dependency of forward reaction rate constant - _j surface concentration of adsorbed species j - electrode area per unit length - residence time of fluid in recycle loop     

Effects of temperature and pressure gradients on catalyst pellet effectiveness factors—I

The dusty gas model is used to establish the effects of temperature and pressure gradients on catalyst pellet effectiveness factors for reaction systems in which species molecular weights and transport coefficients are indistinguishable and Σν_i = 0. For this class of reactions, the total molar flux of species i is shown to be expressible simply as N_i = ?cD ??_i in terms of the molar concentration, the Bosanguet diffusivity, and the mole fraction gradient. The effects of temperature and pressure gradients are reflected only in variations in molar concentration and diffusivity. Furthermore, the temperature-pressure relationship is shown to be given by the thermal transpiration equation for a pure gas.Typical numerical results are reported for first order reactions in spherical pellets under diffusive conditions ranging from the Knudsen through the bulk diffusion regimes.The variation in diffusion regime is shown to be controlled by an additional parameter α, the Knudsen to bulk diffusion ratio. Comparisons are made with the classical Weisz-Hicks nonisothermal pellet solutions based on N_i = ?D_eff?c_i. For highly exothermic reactions, effectiveness factors are 18% lower in the Knudsen regime and 30% higher in the bulk diffusion regime than are the Weisz—Hicks values. For highly endothermic reactions with a significant diffusion limitation, the effectiveness factors are 30% lower than the Weisz—Hicks values.The classical Damköehler relationship for pellet temperature rise is shown to apply in the Knudsen regime, with the maximum dimensionless center temperature given by (1 + β), where β is the heat of reaction parameter. This temperature is accompanied by a maximum dimensionless center pressure of (1 + β)^{$\text{1}\text{2}$}.In the bulk diffusion regime, the maximum center temperature is shown to be increased by the additional term β²/4. This additional temperature rise accounts for the 42% increased bulk diffusion effectiveness for highly exothermic reactions.     

Electrodissolution of aluminium thin film microband electrodes

This paper discusses the electrodissolution of aluminium thin films as microband electrodes (length = 5 × 10^–3 m) in terms of mass transfer determined by voltammetry and a.c.-impedance techniques as a function of bandwidth (20 to 2000 nm) in 0.1m NaOH solution. The anodic polarization curves of the aluminium microband electrodes show that current density is enhanced with decreasing bandwidth. The ac impedance response suggests that a steady-state diffusion layer appears the more markedly, the smaller the bandwidth. The anodic polarization curves are analysed on the basis of the combined Butler-Volmer high field approximation and the semi-cylindrical diffusion field approximation. As a result of the analysis, the electrodissolution proceeds by a mixed kinetic-mass transfer controlled reaction. The analysis also makes it possible to distinguish the semi-cylindrical diffusive mass transfer contribution to the electrodissolution from the kinetic contribution, i.e., mass transfer index linearly diminishes with decreasing bandwidth. The increased current density is attributable to the decreased mass transfer contribution, i.e., the more predominant semi-cylindrical diffusive mass transfer as compared to laminar diffusive mass transfer.Nomenclature k _a anodic kinetic constant - k _c cathodic kinetic constant - F Faraday constant - _s kinetic transfer coefficient for anodic reaction - _c kinetic transfer coefficient for cathodic reaction - c ^s surface concentration - V anodic polarization - D diffusion coefficient - d diffusion layer thickness - z number of electrons transferred - l length of microband electrode - w bandwidth of microband electrode - r radius of cylinder     

Adsorption and diffusion of nitrogen and oxygen in a carbon molecular sieve

The adsorption and diffusion of N₂ and O₂ in two samples of carbon molecular sieve (Bergbau-Forschung) have been studied by gravimetric and chromatographic methods. The uptake curves (for N₂) show a rapid initial uptake followed by a much slower approach to equilibrium in the long time region. This behaviour can be accounted for by a dual stage (micropore/macropore) diffusion model. Diffusion of oxygen was too fast to be measured accurately by the gravimetric method although an approximate estimate may be obtained from the initial uptake rate. Over the range of the measurements (0–1 atm), the equilibrium isotherms for both O₂ and N₂ are almost linear and there is very little difference between them. Micropore diffusivities derived from the second moments of the chromotographic response peaks are in reasonable agreement with the values derived from the uptake curves [(D_c/r_c²)_O2 ≈ 3.7 × 10⁻³ s⁻¹; (D_c/r_c²)_N2 ≈ 1.2 × 10⁻⁴ s⁻¹ at 303 K], indicating a kinetic selectivity (D_O2/D_N2) of about 30. This is close to the kinetic ratio reported by Knoblauch (for a Bergbau-Forschung sieve) although our values of D_c/r_c² are considerably larger.     

The oxidation of Cu(I) by oxygen gas in concentrated NaCl solutions: A kinetic study

The oxidation of Cu(I) by oxygen gas in concentrated NaCl solutions was studied in a standard stirred reactor. Each test was carried out at constant pH and under stable hydrodynamic conditions. The effect of stirring speed, solution volume, temperature, pH, oxygen partial pressure, chloride and copper ion concentration of the solution were evaluated.The oxidation of Cu(I) followed linear kinetics up to 85% of the copper ions in the cupric state. The experimental results were interpreted by using the film theory of mass transfer with chemical reaction. In the linear range, the chemical reaction was found to be first order with respect to oxygen concentration. The kinetic constant is given. The flux equation was specific of a fast reaction regime and the overall reaction rate was independent of the mass transfer coefficient k_L up to a k_L-value of about 10^?2 cm/sec. A parallel study of the oxidation of SO₃^2? by oxygen gas was carried out in the same reactor. The effects of stirring speed and solution volume were similar for both systems.     

Numerical models for reactions catalysed by homogeneous mediators: the case of Fenton's reagent

A numerical model has been developed to describe the behaviour of a batch reactor in which Fenton's reagent is used for hydroxylating aromatic hydrocarbons under conditions of electrochemical regeneration. The test reaction considered is the conversion of benzene into phenol. Comparison is made with previously published experimental results.Nomenclature A electrode area, m² - a ₁ parameter defined by Equation 21 - C _i concentration of species, i, in the bulk solution, mol m^–3 - c _i local concentration of species, i, in the diffusion layer, mol m^–3 - K _i effective mass-transfer coefficient, m s^–1 - k _j rate constant of reaction j - R _j rate of reaction j, mol m^–3 s^–1 - r _i rate of change of concentration of species i due to chemical reaction, mol m^–3 s^–1 - t time, s - V reactor volume, m³ - x distance from the cathode surface, m - x ^* maximum thickness of the diffusion layer, m - period of diffusion layer renewal, s Subscrpts 1 oxygen - 2 Fe³⁺ - 3 hydrogen peroxide - 4 Fe²⁺ - 5 benzene - 6 phenol - 7 biphenyl This paper was presented at the meeting on Electroorganic Process Engineering held in Perpignan, France, 19–20 September 1985.     

Electrochemical impedance spectroscopic investigation of CO2 reduction on polyaniline in methanol

The ac response of polyaniline thin films on platinum electrodes was measured at different dc potentials during the CO₂ reduction in methanol/LiClO₄ electrolyte with a small amount of 0.5 M H₂SO₄. The complex capacitance curves were simulated and the data obtained were used to calculate kinetic parameters, based on the assumption that the thermodynamic potential E₀ is in the region of −0.2-−0.1 V versus saturated calomel electrode (SCE). With E₀=−0.2 V versus SCE and β=0.6, a j₀ value of ca. 10⁻⁴ A cm⁻² was found for the electroreduction of CO₂ on the polyaniline electrode.     

A direct graphical analysis of polarographic kinetic currents

The linearity of logarithmic plots of i_k/(i_d ? i_k) vs C_j for polarographic kinetic currents allows direct graphical analysis of the kinetics of the prior chemical step with respect to species j without the need for reference to Koutecky's table. Results obtained in this way compare favorably with those secured using the conventional Koutecky analysis.     

A fractal-like kinetics equation to calculate landfill methane production

Landfill appears as a convenient choice to get rid of municipal solid waste while providing energy, due to methane generated through anaerobic fermentation. However, without capture and treatment landfill gas is considered an important source of atmospheric methane. The control and use of this gas require knowledge of both, current yield and long-term accumulative production. These values are usually calculated with mathematical expressions that consider 100% of conversion, and homogeneous chemical reactivity inside the fill. Nevertheless, fermentation in landfills is erratic and spatially heterogeneous. This work introduces a fractal-like chemical kinetics equation to calculate methane generation rate from landfill, Q_CH4 (m³/year), in the way: where fermentable wastes are partitioned in readily, moderately and slowly biodegradable categories, L₀ is the potential of methane yield of refuse (m³/tonne under standard conditions), d_s is the solid-phase fracton dimension, k_i is the reaction kinetics constant of waste category i (year⁻¹), and t_j is the time from the year of burying j (year), C_ij⁰ (kg/tonne) and M_ij (kg) are the initial concentration and the mass of waste category i landfilled in year j, respectively. The idea behind this equation is that methane production kinetics is limited by the diffusion of hydrolyzed substrate into a heterogeneous solid-phase towards discrete areas, where methanogenesis occurs. A virtual study for a hypothetical case is developed. The predictions from this fractal approach are contrasted with those coming from two equations broadly used in the industrial work. The fractal-like kinetics equation represents better the heterogeneous nature of the fermentation in landfills.     

Fluoride removal from water by meixnerite and its calcination product

Fluoride removal from water by a mechanochemically synthesized anion clay (meixnerite) and its calcination product was studied at initial fluoride:meixnerite molar ratios (F_I:meix) of 0.1 to 2.0 — the theoretical fluoride uptake limit for meixnerite. Fluoride removal efficiency of calcined meixnerite was higher than uncalcined meixnerite at the same F_I:meix, and the difference increased as the initial fluoride concentration increased. For the sorption runs performed at F_I:meix = 2.0, 29% and 52% of the uptake capacity were attained for the uncalcined and calcined meixnerites, respectively. Analysis of sorption reaction rate data indicates that fluoride diffusion from solution to intraparticle active sites and its chemical sorption on active sites are important mechanisms in the uptake for both meixnerites, and the intraparticle fluoride diffusion in uncalcined meixnerite was slower than in calcined meixnerite. Moreover, XRD analyses indicate that secondary fluoride-containing phases (nordstrandite and sellaite) precipitated at high initial fluoride concentrations. When fluoride precipitates did not form at lower F_I:meix (< 0.6), the higher fluoride uptake by calcined meixnerite is promoted by greater availability of F⁻ ions to the meixnerite interlayers since the interlayers were generated during reaction of the F-containing solution with the calcined material. Thus some F⁻ did not have to diffuse into the interlayers to replace existing OH⁻ ions as it did for the uncalcined meixnerite. At F_I:meix ≥ 0.6, precipitation of F-bearing nordstrandite also contributes to calcined meixnerite's improved ability to remove fluoride.