Stefan solution

This is an explicit solution of the Stefan-Maxwell equations for the diffusion fluxes. The species flux vectors are then given by... 

Low-PressureAlulticomponent Mixtures These methods are outlined in Table 5-17. Stefan-MaxweU equations were discussed earlier. Smith-Taylor compared various methods for predicting multi-component diffusion rates and found that Eq. (5-204) was superior among the effective diffusivity approaches, though none is very good. They so found that hnearized and exact solutions are roughly equivalent and accurate. 

Pinto-Graham Pinto and Graham studied multicomponent diffusion in electrolyte solutions. They focused on the Stefan-Maxwell equations and corrected for solvation effects. They achieved excellent results for 1-1 electrolytes in water at 25°C up to concentrations of 4M. 

The most common states of a pure substance are solid, liquid, or gas (vapor), state property See state function. state symbol A symbol (abbreviation) denoting the state of a species. Examples s (solid) I (liquid) g (gas) aq (aqueous solution), statistical entropy The entropy calculated from statistical thermodynamics S = k In W. statistical thermodynamics The interpretation of the laws of thermodynamics in terms of the behavior of large numbers of atoms and molecules, steady-state approximation The assumption that the net rate of formation of reaction intermediates is 0. Stefan-Boltzmann law The total intensity of radiation emitted by a heated black body is proportional to the fourth power of the absolute temperature, stereoisomers Isomers in which atoms have the same partners arranged differently in space, stereoregular polymer A polymer in which each unit or pair of repeating units has the same relative orientation, steric factor (P) An empirical factor that takes into account the steric requirement of a reaction, steric requirement A constraint on an elementary reaction in which the successful collision of two molecules depends on their relative orientation. 

In Madejski s full model,l401 solidification of melt droplets is formulated using the solution of analogous Stefan problem. Assuming a disk shape for both liquid and solid layers, the flattening ratio is derived from the numerical results of the solidification model for large Reynolds and Weber numbers ... 

To use these models, the freezing constant, U, must be determined. One choice is the solution of the Stefan problem of solidification, as described by Madejski 401 ... 

To rationalize the isothermal assumption, Dykhuizen 39() discussed two related physical phenomena. First, heat may be drawn out of the substrate from an area that is much larger than that covered by asplat. Thus, the 1 -D assumption in the Stefan problem becomes invalid, and a solution of multidimensional heat conduction may make the interface between a splat and substrate closer to isothermal. Second, the contact resistance at the interface is deemed to be the largest thermal resistance retarding heat removal from the splat. If this resistance does not vary much with substrate material, splat solidification should be independent of substrate thermal properties. Either of the phenomena would result in a heat-transfer rate that is less dependent on the substrate properties, but not as high as that calculated by Madej ski based on the... 

Divisek et al. presented a similar two-phase, two-dimensional model of DMFC. Two-phase flow and capillary effects in backing layers were considered using a quantitatively different but qualitatively similar function of capillary pressure vs liquid saturation. In practice, this capillary pressure function must be experimentally obtained for realistic DMFC backing materials in a methanol solution. Note that methanol in the anode solution significantly alters the interfacial tension characteristics. In addition, Divisek et al. developed detailed, multistep reaction models for both ORR and methanol oxidation as well as used the Stefan—Maxwell formulation for gas diffusion. Murgia et al. described a one-dimensional, two-phase, multicomponent steady-state model based on phenomenological transport equations for the catalyst layer, diffusion layer, and polymer membrane for a liquid-feed DMFC. 

Stefan, M.I. and Bolton, J.R. Mechanism of thedegradationof 1,4-dioxane in dilute aqueous solution using the UV/hydrogen peroxide process, Environ. Sci. Technol, 32(11) 1588-1595,1998. 

So far, the selectivity of these membranes was tested only with regard to solutions of salt in water. There are indications however, from various sources, like the work of Peter and Stefan (O, that these membranes may be valuable for other separations. 

In the formulation and solution of conservation equations, we tend to prefer the direct evaluation of the diffusion velocities as discussed in the previous section. However, it is worthwhile to note that the Stefan-Maxwell equations provide a viable alternative. At each point in a flow field one could solve the system of equations (Eq. 3.105) to determine the diffusion-velocity vector. Solution of this linear system is equivalent to determining the ordinary multicomponent diffusion coefficients, which, in this formulation, do not need to be evaluated. 

These constraints must be satisfied in the solution of the Stefan-Maxwell equations. At a point within a chemically reacting flow simulation, the usual situation is that the diffusion velocities must be evaluated in terms of the diffusion coefficients and the local concentration, temperature, and pressure fields. One straightforward approach is to solve only K — 1 of Eqs. 3.105, with the X4h equation being replaced with a statement of the constraint. For... 

To deal with this more complex problem, we follow Sekerka et al. [2] and Sekerka and Wang [3] and first establish a general analysis that allows for these changes of volume. The previous melting problem was solved by first obtaining independent solutions to the diffusion equation in each phase and then coupling them via the Stefan flux condition at the interface, similar approach can be employed for the present problem. To accomplish this, it is necessary to identify suitable frames for analyzing the diffusion in each phase and then to find the relations between them necessary to construct the Stefan condition. 

First, solutions of the diffusion equation in the a and j3 phases (in the F - and F -frame, respectively) are found that match the boundary and initial conditions and then the Stefan condition is invoked. The solutions are of the error-function type and are given by... 

A complete general solution therefore requires solving for temperature and concentration fields in both phases which satisfy all boundary conditions as well as the two coupled Stefan conditions. Solving this problem is a challenging task [4, 5] however, an analysis of the concentration spike under certain simplifying conditions when v is known is given in Section 22.1.1. 

We now determine rjx by solving for c% and invoking the Stefan condition at the interface. The diffusion equation was scaled and integrated in Cartesian coordinates in Section 4.2,2 with the solution given by Eq. 4,28. When this solution is matched to the present boundary conditions,... 

This particular problem was first studied by Stefan (S10), and the general class of solutions of the diffusion equation subject to a free boundary condition, therefore, are sometimes called Stefan problems. An analogous problem in the freezing of moist soils was previously studied, however, by Lame and Clapeyron (LI), and in the Russian literature these problems are sometimes given the rather lengthy soubriquet of Larnd-Clapeyron-Stefan problems. 

Recently Ruoff (Rll) has rederived the Stefan and Neumann solutions using the Boltzmann transformation. 

Stefan gave an exact solution for the constant-velocity melting of a semi-infinite slab initially at the fusion temperature. This was extended by Pekeris and Slichter (P2) to freezing on a cylinder of arbitrary surface temperature and Kreith and Romie (K6) to constant-velocity melting of cylinders and spheres by a perturbation method, in which the temperature is assumed to be expressible in terms of a convergent series of unknown functions. To make the method clear, consider the freezing of an infinite cylinder of liquid, of radius r0, at constant surface heat flux. For this geometry the heat equation is... 

Solute-solute Interactions may affect the diffusion rates In the fluid phase, the solid phase, or both. Toor (26) has used the Stefan-Maxwell equations for steady state mass transfer In multicomponent systems to show that, in the extreme, four different types of diffusion may occur (1) diffusion barrier, where the rate of diffusion of a component Is zero even though Its gradient Is not zero (2) osmotic diffusion, where the diffusion rate of a component Is not zero even though the gradient Is zero (3) reverse diffusion, where diffusion occurs against the concentration gradient and, (4) normal diffusion, where diffusion occurs In the direction of the gradient. While such extreme effects are not apparent in this system, it is evident that the adsorption rate of phenol is decreased by dodecyl benzene sulfonate, and that of dodecyl benzene sulfonate increased by phenol. 

Blackburn AC, Doe WF, Buffinton GD (1998) Salicylate hydroxylation as an indicator of OH radical generation in dextran sulfate-induced colitis. Free Rad Biol Med 25 305-313 Bonifacic M, Stefanic I, Hug GL, Armstrong DA, Asmus K-D (1998) Glycine decarboxylation The free radical mechanism. J Am Chem Soc 120 9930-9940 Bonifacic M, Armstrong DA, Stefanic I, Asmus K-D (2003) Kinetic isotope effect for hydrogen abstraction by OH radicals from normal and carbon-deuterated ethyl alcohol in aqueous solution. J Phys Chem B 107 7268-7276... 

Bonifacic M, Armstrong DA, Stefanic I, Asmus K-D (2003) Kinetic isotope effect for hydrogen abstraction by OH radicals from normal and carbon-deuterated ethyl alcohol in aqueous solution. J Phys Chem B 107 7268-7276... 

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