Product Layer Model

By allowing the increment to approach zero (Ar 0),we obtain the differential equation 

Taking into account that y = dcA/dr and integrating Equation 8.30 once more, the concentration profile of A is obtained accordingly  

According to Equation 8.30, the concentration gradient is obtained firom 

This flux is equal to the flux through the gas film according to Equation 8.19  

Inserting the concentration onto the particle surface c. Equation 8.32, yields 

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A solid product or an inert residue is formed and the reaction rate is slow (no influence of diffusion through boundary gas and solid product layer) (model 1 in Figure 4.6.1). 

Komatsu [478] has put forward the hypothesis that reaction in many powder mixtures is initiated only at interparticle contact and that product formation occurs by diffusion through these contact zones. Here, one of the participating reactants is not covered with a coherent product layer. Quantitative consideration of this model leads to a modified Jander equation. 

The inclusion of radiative heat transfer effects can be accommodated by the stagnant layer model. However, this can only be done if a priori we can prescribe or calculate these effects. The complications of radiative heat transfer in flames is illustrated in Figure 9.12. This illustration is only schematic and does not represent the spectral and continuum effects fully. A more complete overview on radiative heat transfer in flame can be found in Tien, Lee and Stretton [12]. In Figure 9.12, the heat fluxes are presented as incident (to a sensor at T,, ) and absorbed (at TV) at the surface. Any attempt to discriminate further for the radiant heating would prove tedious and pedantic. It should be clear from heat transfer principles that we have effects of surface and gas phase radiative emittance, reflectance, absorptance and transmittance. These are complicated by the spectral character of the radiation, the soot and combustion product temperature and concentration distributions, and the decomposition of the surface. Reasonable approximations that serve to simplify are ... 

Charbeneau, R. J., Wanakule, N., Chiang, C. Y., Nevin, J. R, and Klein, C. L., 1989, A Two-Layer Model to Simulate Floating Free Product Recovery Formulation and Applications In Proceedings of the National Water Well /Association and American Petroleum Institute Conference on Petroleum Hydrocarbons and Organic Chemicals in Ground Water Prevention, Detection and Restoration, November, pp. 333-345. 

The present research has treated important parts of the modeling of combustion and NOx formation in a biomass grate furnace. All parts resulted in useful approaches. For all these approaches successful first steps were taken. Currently, more research is underway to obtain improved results NH3 production is measured in the grid reactor with the tunable diode laser, detailed kinetics will be attached to the front propagation model, including the measured NH3 release functionalities, and for the turbulent combustion model heat losses are taken into account. In addition, the fuel layer model has to be coupled to the turbulent combustion model in the furnace. 

To model this, Duncan-Hewitt and Thompson [50] developed a four-layer model for a transverse-shear mode acoustic wave sensor with one face immersed in a liquid, comprised of a solid substrate (quartz/electrode) layer, an ordered surface-adjacent layer, a thin transition layer, and the bulk liquid layer. The ordered surface-adjacent layer was assumed to be more structured than the bulk, with a greater density and viscosity. For the transition layer, based on an expansion of the analysis of Tolstoi [3] and then Blake [12], the authors developed a model based on the nucleation of vacancies in the layer caused by shear stress in the liquid. The aim of this work was to explore the concept of graded surface and liquid properties, as well as their effect on observable boundary conditions. They calculated the hrst-order rate of deformation, as the product of the rate constant of densities and the concentration of vacancies in the liquid. 

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 ... 

The parabolic-rate law for the growth of thick product layers on metals was first reported by Tammann (1920), and a theoretical interpretation in terms of ambipolar diffusion of reactants through the product layer was advanced later by Wagner (1936, 1975). Wagner s model can be described qualitatively as follows when a metal is... 

The maximum sulfation observed is considerably smaller than the theoretical limit of 59 percent. This observation, together with the decrease in rate constant Kg with extent of sulfation, can be explained by models which assume that the reaction is retarded by a product layer of CaSO of low porosity through which the SO2 must diffuse in order to reach the unreacted CaO. Such a hypothesis has formed the basis for the development of several models to quantitatively fit sulfation data, of which the grain model (11, 12) and the pore-plugging model (13, 14) are the most notable. 

In the grain model, it is assumed that the CaO consists of spherical grains of uniform size distributed in a porous matrix. The rate of reaction is controlled by the diffusion of SO2 through the porous matrix and the product CaSO layer formed on each grain (11). Allowance can be made for a finite rate of the CaO/SC reaction (12). The models have been found to describe experimental data for many limestones (13) by adjusting the constants in the model, most notably the diffusivity through the product layer. 

Passivation potential — Figure 2. Evaluation of XPS data on the chemical structure of the passive layer on Fe formed for 300 s in 1M NaOH as a function of potential with a two-layer model Fe(II)/Fe(III). Insert shows the polarization curve with oxidation of Fe(II) to Fe(III) at the Flade potential EP2, indication of soluble corrosion products Fe2+ and Fe3+, and passivation potential EPi in alkaline solution [i, iii]... 

While providing a simple method for analyzing the redistribution of energy in the combustion wave, the models discussed in the previous section do not account for the local structural features of the reaction medium. Microstructural models account for details such as reactant particle size and distribution, product layer thickness, etc., and correlate them with the characteristics of combustion (e.g., U,T,). 

The equations used to describe the combustion wave propagation for microstructural models are similar to those in Section IV,A [see Eq. (6)]. However, the kinetics of heat release, 4>h may be controlled by phenomena other than reaction kinetics, such as diffusion through a product layer or melting and spreading of reactants. Since these phenomena often have Arrhenius-type dependences [e.g., for diffusion, 2)=9)o exp(— d// T)], microstructural models have similar temperature dependences as those obtained in Section IV,A. Let us consider, for example, the dependence of velocity, U, on the reactant particle size, d, a parameter of medium heterogeneity ... 

The first microstructural models were developed independently and essentially simultaneously (Aldushin et al 1972a,b Hardt and Phung, 1973 Aldushin and Khaikin, 1974). For these models, the elementary reaction cell, which accounts for the details of the microstructure, consists of alternating lamellae of the two reactants (A and B), which diffuse through a product layer (C), to react (see Fig. 20a). Assuming that the particles are flat allows one to neglect the change in reac-... 

For the next two types of theoretical models, the elementary reaction cell consists of a spherical particle of one reactant surrounded by a melt of the other reactant. In the first case, the product layers (C) grow on the surface of the more refractory particles (B) due to diffusion of atoms from the melt phase (A) through the product layer (see Fig. 20b). At a given temperature, the concentrations at the interphase boundaries are determined from the phase diagram of the system. Numerical calculations by Nekrasov et al. (1981, 1993) have shown satisfactory agreement with experimental results for a variety of systems. 

Hedgecock I. M. and Pirrone N. (2001) Mercury and photochemistry in the marine boundary layer-modelling studies suggest the in situ production of reactive phase mercury. Atmos. Environ. 35, 3055-3062. 

A m, the apparent constant, is the product of an intrinsic constant (a constant valid for a hypothetical uncharged surface) and a Boltzmann factor. / is the surface potential, F the Faraday constant, and AZ the change in the charge number of the surface species of the reaction for which the equilibrium constant is defined (in this case AZ = -bl). The intrinsic constant is experimentally accessible by extrapolating experimental data to the surface charge where op = 0 and where l/ = 0. The correction, as given above, assumes the classical diffuse double-layer model (a planar surface and a diffuse layer of counterions). 

This model is applicable to the reactions of nonporous pellets and to porous pellets when the global rate is controlled by pore diffusion. Reaction is limited to a surface separating the solid reactant at the core of the pellet surrounded by a porous layer of solid product. It occurs initially on the external surface of the pellet, and the thickness of the product layer increases as the reaction proceeds, as illustrated in Fig. 1. The global reaction rate is determined by three resistances— mass transfer from bulk gas to particle surface, diffusion... 

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