Stefan velocity

The surface-normal component of the diffusion velocity is Vkbulk fluid velocity (Stefan velocity) at the surface is given as... 

For surfaces where solid films are deposited or removed, there is a net mass exchange with the gas. At each surface i (upper and lower, if applicable) the net mass exchange is determined from the Stefan velocity as... 

In steady state, for surfaces where the chemistry is only catalytic, the Stefan velocity is zero because there is no net mass exchange. However, it should be noted that there may be net mass exchange during a transient process such as a catalytic ignition [323] as coverage on the surface may vary. 

The expression for the Stefan velocity is easily obtained from the interfacial mass balance (Eq. 11.123) by summing over all Kg species and noting that the mass fractions must sum to one,... 

Fig. 17.15 The top panel shows the transient surface-state composition during catalytic ignition on a long time scale. The lower panel shows the transient response of the Stefan velocity and the pressure-curvature eigenvalue on a very short time scale during the ignition transient. The zero point for the abscissa scales is arbitrary.

For all steady-state applications, the Stefan velocity is identically zero (there is no etching or deposition in steady-state catalytic combustion). Given the ID dimensionality of the stagnation-flow problem, a full multicomponent transport approach for the diffusion velocities is computationally manageable ... 

If the boundary motion is controlled by an independent process, then the boundary motion velocity is independent of diffusion. This can happen if the magma is gradually cooling and crystal growth rate is controlled both by temperature change and mass diffusion. This problem does not have a name. In this case, u depends on time or may be constant. If the dependence of u on time is known, the problem can also be solved. The Stefan problem and the constant-w problem are covered below. 

In the Stefan-Maxwell setting [35,178,435], the diffusion velocities are related implicitly to the field gradients as follows ... 

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... 

In nonreacting, continuum fluid mechanics the fluid velocity normal to a solid wall is zero, which is a no-slip boundary condition. However, if there are chemical reactions at the wall, then the velocity can be nonzero. The so-called Stefan flow velocity occurs... 

No approximation is made when using Eqs. 12.168 and 12.169. Exactly the same mass diffusion velocities and fluxes are obtained in this approach as would be calculated via Eqs. 12.165 and 12.166 (or from the Stefan-Max well approach described next). 

The Stefan-Maxwell equations (12.170 and 12.171) form a system of linear equations that are solved for the K diffusion velocities V. The diffusion velocities obtained from the Stefan-Maxwell approach and by evaluation of the multicomponent Eq. 12.166 are identical. 

The Stefan-Maxwell diffusion velocities are, in general, solved from a set of K (linear) simultaneous equations. Equation 12.170, with the A th equation equation replaced by Eq. 12.171, can be rewritten in matrix form as... 

In this example, the equilibrium concentrations maintained at the interface are functions of the interface temperature, which in turn is a function of time. In addition, the velocity of the interface, v (i.e., rate of solidification), depends simultaneously upon the mass diffusion rates and the rates of heat conduction in the two phases, as may be seen by examining the two Stefan conditions that apply at the interface. For the mass flow the condition is... 

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... 

The velocity at any point uy(6) is determined by the rate of melting at the interface (Fig. 6.67), which is obtained from the Stefan condition or heat balance between conduction and the rate of melting at that interface,... 

For the kth species Y]< and are the mass and mole fractions, respectively, R]< is the net rate of the production due to chemistry and is the molecular weight. In addition the diffusion velocity V is given by the Stefan-Maxwell relation... 

Tables 2.12-2.14 show some values of diffusion coefficients in solids and polymers. In a flow of dilute solution of polymers, the diffusivity tensor is anisotropic and depends on the velocity gradient. The Maxwell-Stefan equation may predict the diffusion in multicomponent mixtures of polymers.

Maxwell-Stefan equations describe steady diffusion flows, assuming that shearing forces for each species are negligible. As there are no velocity gradients assumed, the Maxwell-Stefan equations can be written in the forms of fluxes. For a ternary mixture of components 1, 2, and 3, the flow of component 1 in the z direction is... 

The Maxwell-Stefan equations do not depend on choice of the reference velocity, and therefore they are a proper starting point for other descriptions of multicomponent diffusion. For ideal gas mixtures, diffusivities /, and D u are... 

Maxwell described diffusion by velocity differences, which yield forces from the friction between the molecules of different species. He considered a chemical potential gradient caused by friction, which is proportional to the concentration. The diffusion coefficient of Maxwell-Stefan can be defined as... 

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