Solid phase, diffusion models

By a tiial-and-eiTor procedure, using the value of U(t) and the model eqs. (4.52)-(4.58), the solid-phase diffusion coefficient is found to be 1.3 X 10 12 m2/s. This value is very close to the one given in the study of Meshko el al. The trial-and-error procedure can be done easily. By changing the value of Ds, the model predicts the values of U(t) for each t. The best value of Ds is the one that results in the lowest mean deviation between the experimental and the model values of U(t). In Figure 4.19, the performance of the model is shown. The average error is 3.1%. 

Again following a trial-and-eiror procedure, the solid-phase diffusion coefficient is found to be 1.82 X 10-9 cm2/s. This value is very close to the one given in the study of Choy and McKay. In Figure 4.21, the performance of the model is shown. The average error is 3%. 

In the DGM, the solid phase is modeled as giant dust molecules held motionless in space with which the diffusing gas molecules collide. The constitutive equations governing the diffusion molar flux intensities Nf for both MTPM and DGM are the generalized Maxwell-Stefan equations... 

Continuum models encompass both micro and macro scales and in li-ion models the microscale is governed by the solid phase diffusion equation. The coupling of the microscale and the macroscale variables pose computational limitations. 

The solid-phase diffusion of monoatomic metal (or its oxide) particles to give multiatomic cluster particles. It is assumed that diffusion processes in the solid phase are activated. The rate of diffusion = DnC = C Di where C = Cla is the current amount of reaction centers per one cell of the a size, N is the size of A-atomic cluster, Di = Doexp[- aE/( T)] is the diffusion coefficient for a monoatomic particle, EaX) and Do = vexp(A57/ ) are the varying parameters, the energy of activation, and the entropy factor ((v 10 sec" ). Such a model assumes that the coexistence of two or more separate particles in one cell is not possible because they immediately form a single cluster. In the isotropic medium the diffusing particle with a corresponding probability can move in one of the 26 directions. 

A mathematical model describing the processes of AC charging-discharge with account for EDL charging, intercalation of hydrogen into carbon, and its nonsteady-state solid-phase diffusion, electrode kinetics, ion transport over the electrode thickness, and characteristics of its porous structure was developed and confirmed experimentally. It is shown that the maximum path of diffusion of hydrogen atoms all other conditions being equal is inversely proportional to the hydrophilic specific surface area of the electrode. 

For proper design and simulation of HDT reactors, kinetic and reactor modeling are aspects that need to be deeply studied however, this is not a trivial task due to the numerous physical and chemical processes that occur simultaneously in the reactor phase equilibrium, mass transfer of reactants and products between the gas-liquid-solid phases, diffusion inside the catalyst particle, a complex reaction network, and catalyst deactivation. Ideally, the contribution of all these events must be coupled into a robust reactor performance model. The level of sophistication of the model is generally defined based upon the pursued objectives and prediction capability [4]. 

Let us proceed to model the system using one of our formulations, the compartmental model. To do this we assume the liquid to be well mixed witii a uniform concentrahon of This is not an unreasonable assmnption as the size of the pool is quite small, typically 50 to 200 pm in width. The solid phase, on the other hand, caimot be considered uniform because solid-phase diffusivities are several orders of magnitude smaller than those prevailing in the liquid (see Section 3.1 in the next chapter). Concentrations... 

In this approach, originally developed by Fuller et al. [18, 53] based on the porous electrode theory [42], the active material is assumed to consist of spherical particles with a specific size, and solid phase diffusion in the radial direction is assumed to be the predominant mode of transport. The electrolyte phase concentration (Cg) and the potentials (4>s,4>e) are assumed to vary along the principal (i.e., thickness) direction only, and are henceforth referred to as the x direction. In other words, this model implicitly considers two length scales (1D + 1D), that is, the r direction inside the spherical particle and the x direction along the thickness. All other equations described earlier continue to remain valid except the solid phase diffusion. Equation 25.19 and the corresponding boundary/initial conditions. The solid phase diffusion equation now takes the following form ... 

Various pc electrode models have been tested.827 Using the independent diffuse layer electrode model74,262 the value of E n = -0.88 V (SCE) can be simulated for Cd + Pb alloys with 63% Pb if bulk and surface compositions coincide. However, large deviations of calculated and experimental C,E curves are observed at a 0. Better correspondence between experimental and calculated C,E curves was obtained with the common diffuse-layer electrode model,262 if the Pb percentage in the solid phase is taken as 20%. However, the calculated C, at a Ois noticeably lower than the experimental one. It has been concluded that Pb is the surface-active component in Cd + Pb alloys, but there are noticeable deviations from electrical double-layer models for composite electrodes.827... 

Van Orman J, Saal A, Bourdon B, Hauri E (2002a) A new model for U-series isotope fractionation during igneous proeesses with finite diffusion and multiple solid phases. EOS Trans, Am Geophys Union 83(47) Fall Meet Suppl Abstract V71C-02... 

The theory on the level of the electrode and on the electrochemical cell is sufficiently advanced [4-7]. In this connection, it is necessary to mention the works of J.Newman and R.White s group [8-12], In the majority of publications, the macroscopical approach is used. The authors take into account the transport process and material balance within the system in a proper way. The analysis of the flows in the porous matrix or in the cell takes generally into consideration the diffusion, migration and convection processes. While computing transport processes in the concentrated electrolytes the Stefan-Maxwell equations are used. To calculate electron transfer in a solid phase the Ohm s law in its differential form is used. The electrochemical transformations within the electrodes are described by the Batler-Volmer equation. The internal surface of the electrode, where electrochemical process runs, is frequently presented as a certain function of the porosity or as a certain state of the reagents transformation. To describe this function, various modeling or empirical equations are offered, and they... 

The terms Jga and Jsa are the diffusive fluxes of species a in the gas and solid phases, respectively. Note that in addition to molecular-scale diffusion, these terms include dispersion due to particle-scale turbulence. The latter is usually modeled by introducing a gradient-diffusion model with an effective diffusivity along the lines of Eqs. (149) and (151). Thus, for large particle Reynolds numbers the molecular-scale contribution will be negligible. The term Ma is the... 

While first-order models have been used widely to describe the kinetics of solid phase sorption/desorption processes, a number of other models have been employed. These include various ordered equations such as zero-order, second-order, fractional-order, Elovich, power function or fractional power, and parabolic diffusion models. A brief discussion of these models will be provided the final forms of the equations are given in Table 2. 

The Elovich model was originally developed to describe the kinetics of heterogeneous chemisorption of gases on solid surfaces [117]. It describes a number of reaction mechanisms including bulk and surface diffusion, as well as activation and deactivation of catalytic surfaces. In solid phase chemistry, the Elovich model has been used to describe the kinetics of sorption/desorption of various chemicals on solid phases [23]. It can be expressed as [118] ... 

Equation (57) is empirical, except for the case where v = 0.5, then Eq. (57) is similar to the parabolic diffusion model. Equation (57) and various modified forms have been used by a number of researchers to describe the kinetics of solid phase sorption/desorption and chemical transformation processes [25, 121-122]. 

Simulation and predictive modeling of contaminant transport in the environment are only as good as the data input used in these models. Field methods differ from laboratory methods in that an increase in the scale of measurement relative to most laboratory methods is involved. Determination of transport parameters (i. e., transmission coefficients) must also use actual contaminant chemical species and field solid phase samples if realistic values are to be specified for the transport models. The choice of type of test, e.g., leaching cells and diffusion tests, depends on personal preference and availability of material. No test is significantly better than another. Most of the tests for diffusion evaluation are flawed to a certain extent. 

If the transport process is rate-determining, the rate is controlled by the diffusion coefficient of the migrating species. There are several models that describe diffusion-controlled processes. A useful model has been proposed for a reaction occurring at the interface between two solid phases A and B [290]. This model can work for both solids and compressed liquids because it doesn t take into account the crystalline environment but only the diffusion coefficient. This model was initially developed for planar interface reactions, and then it was applied by lander [291] to powdered compacts. The starting point is the so-called parabolic law, describing the bulk-diffusion-controlled growth of a product layer in a unidirectional process, occurring on a planar interface where the reaction surface remains constant ... 

That this was known in industry at least 15 years earlier is one of the unfortunate discrepancies between academic research and commercial industrial research and development. Not all that is known is necessarily published. This realization subsequently lead to the development of both solid phase and gas phase linear driving force models that each provide very good representations of measured data without the excess labor involved with the diffusion-based models. For trace systems there are quite a few analytical solutions that are available and quite tractable for both design work and the analysis of adsorption column performance. 

Because the Adler model is time dependent, it allows prediction of the impedance as well as the corresponding gaseous and solid-state concentration profiles within the electrode as a function of time. Under zero-bias conditions, the model predicts that the measured impedance can be expressed as a sum of electrolyte resistance (Aeiectroiyte), electrochemical kinetic impedances at the current collector and electrolyte interfaces (Zinterfaces), and a chemical impedance (Zchem) which is a convolution of contributions from chemical processes including oxygen absorption. solid-state diffusion, and gas-phase diffusion inside and outside the electrode. 

A model of such structures has been proposed that captures transport phenomena of both substrates and redox cosubstrate species within a composite biocatalytic electrode.The model is based on macrohomo-geneous and thin-film theories for porous electrodes and accounts for Michaelis—Menton enzyme kinetics and one-dimensional diffusion of multiple species through a porous structure defined as a mesh of tubular fibers. In addition to the solid and aqueous phases, the model also allows for the presence of a gas phase (of uniformly contiguous morphology), as shown in Figure 11, allowing the treatment of high-rate gas-phase reactant transport into the electrode. 

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