Diffusion equation, chemical species

The diffusion of chemical species is the predominant mechanism driving corrosion in neutral solutions. After water diffusion into the glass network, hydration of alkaline oxides present in all glass formulations (even the most resistant) leads to the diffusion of Na" and OH ions towards the surface and the aqueous medium (Equation [12.66]). Hydroxide ions may then lead to the hydrolysis of siloxane bonds (etching) without being consumed, as shown in Equations [12.67] and [12.68] (Ishai, 1975). This is an autocatalytic process, as the rate of dissolution of the glass increases with time. 

The computational model assumes the mixing process of one species dissolved in water without chemical reactions occurring during the mixing and thus without the reaction term included in the species equation. Therefore, only the steady-state convection-diffusion equation for species A has to be considered as... 

In addition to the Scheibel method discussed previously (Equations 5.26 and 5.27), another semiempirical model for estimating diffusion coefficients in organic liquids developed by Hayduk and Minhas (1982) is available. Though this model has similar functionality to the other liquid diffusivity estimation methods described above, this formula is specifically for diffusion of chemical species in paraffin solutions... 

Thus far we have considered systems where stirring ensured homogeneity witliin tire medium. If molecular diffusion is tire only mechanism for mixing tire chemical species tlien one must adopt a local description where time-dependent concentrations, c r,f), are defined at each point r in space and tire evolution of tliese local concentrations is given by a reaction-diffusion equation... 

In the case of systems containing ionic liquids, components and chemical species have to be differentiated. The methanol/[BMIM][PF6] system, for example, consists of two components (methanol and [BMIM][PFg]) but - on the assumption that [BMIM][PFg] is completely dissociated - three chemical species (methanol, [BMIM] and [PFg] ). If [BMIM][PFg] is not completely dissociated, one has a fourth species, the undissociated [BMIM][PFg]. From this it follows that the diffusive transport can be described with three and four flux equations, respectively. The fluxes of [BMIM] ... 

Pollard and Newman" have also studied CVD near an infinite rotating disk, and the equations we solve are essentially the ones stated in their paper. Since predicting details of the chemical kinetic behavior is a main objective here, the system now includes a species conservation equation for each species that occurs in the gas phase. These equations account for convective and diffusive transport of species as well as their production and consumption by chemical reaction. The equations stated below are given in dimensional form since there is little generalization that can be achieved once large chemical reaction mechanisms are incorporated. 

For problems involving gradients in chemical species, the convection-diffusion equations for the species are also solved, usually for N— 1 species with the Nth species obtained by forcing the mass fractions to sum to unity. Turbulence can be described by a turbulent diffusivity and a turbulent Schmidt number, Sct, analogous to the heat transfer case. 

The chemical species were treated as passive scalar tracers in the unsteady LBM equations. The reaction was simulated as being mass-transfer limited at low Re — 166, with diffusivities in the ratios DA DB Dc— 1 3 2. The concentration fields shown in Fig. 16 are different for each species due to the different diffusivities. The slow-diffusing species A is transported mainly by convection and regions of high or low concentration correspond to features of the flow field. A more uniform field is seen for the concentration of faster... 

In the special case where all chemical species have the same molecular diffusivity, only one transport equation is often required to describe the conserved scalars. The single conserved scalar can then be expressed in terms... 

For the case of classical Gaussian diffusion 0=0 and, believing r(t)=2 and t=4 relative units, the equality within the framework of the relationship (1) will be obtained. Such equality assumes p= 1, i.e., each contact of reagents molecules results to reaction product formation. Let s assume, that the value p decreases up to 0,05, i.e., only one from 20 contacts of reagents molecules forms a new chemical species. This means the increase t in 20 times and then at r(t)=2 and =80 relative units from the relationship (1) will be obtained 0=4,33. Since 0 is connected with dimension of walk trajectory of reagents molecules dw by the simple equation... 

A mathematically simple case, that occurs frequently in solvent extraction systems, in which the extracting reagent exhibits very low water solubility and is strongly adsorbed at the liquid interface, is illustrated. Even here, the interpretation of experimental extraction kinetic data occurring in a mixed extraction regime usually requires detailed information on the boundary conditions of the diffusion equations (i.e., on the rate at which the chemical species appear and disappear at the interface). 

The numerical jet model [9-11] is based on the numerical solution of the time-dependent, compressible flow conservation equations for total mass, energy, momentum, and chemical species number densities, with appropriate in-flow/outfiow open-boundary conditions and an ideal gas equation of state. In the reactive simulations, multispecies temperature-dependent diffusion and thermal conduction processes [11, 12] are calculated explicitly using central difference approximations and coupled to chemical kinetics and convection using timestep-splitting techniques [13]. Global models for hydrogen [14] and propane chemistry [15] have been used in the 3D, time-dependent reactive jet simulations. Extensive comparisons with laboratory experiments have been reported for non-reactive jets [9, 16] validation of the reactive/diffusive models is discussed in [14]. 

The systems considered here are isothermal and at mechanical equilibrium but open to exchanges of matter. Hydrodynamic motion such as convection are not considered. Inside the volume V of Fig. 8, N chemical species may react and diffuse. The exchanges of matter with the environment are controlled through the boundary conditions maintained on the surface S. It should be emphasized that the consideration of a bounded medium is essential. In an unbounded medium, chemical reactions and diffusion are not coupled in the same way and the convergence in time toward a well-defined and asymptotic state is generally not ensured. Conversely, some regimes that exist in an unbounded medium can only be transient in bounded systems. We approximate diffusion by Fick s law, although this simplification is not essential. As a result, the concentration of chemicals Xt (i = 1,2,..., r with r sN) will obey equations of the form... 

In the previous chapter, several factors which complicate the simple diffusion equation analysis of chemical reactions in solution were discussed rather qualitatively. However, the magnitude of these effects can only be gauged satisfactorily by a detailed physical and mathematical analysis. In particular, the hydrodynamic repulsion and competitive effects have been studied recently by a number of workers. Reactions between ionic species in solutions containing a high concentration of ionic species is a similarly involved subject. These three instances of complications to the diffusion equation all involve aspects of many-body effects. 

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