Steady-state binary diffusion

Steady-State Binary Molecular Diffusion in Porous Solids... 

Describe Fick s model of diffusion in words and equations, and use the model to solve steady-state binary diffusion problems without convection... 

Choose an appropriate reference velocity Vj- f and solve Fick s model for steady-state binary diffusion with convection... 

Example 7.3-2 Steady-state multicomponent diffusion across a thin film In steady-state binary diffusion, we found that the solute s eoncentration varied linearly across a thin film. Will solute concentrations vary linearly in the multieomponent ease What will the flux be Solution By comparison with Eq. 2.2-9, we see that... 

Despite the very restricted circumstances In which these equations properly describe the dynamical behavior, they are the starting point for almost all the extensive literature on the stability of steady states in catalyst pellets. It is therefore Interesting to examine the case of a binary mixture at the opposite limit, where bulk diffusion controls, to see what form the dynamical equations should take in a coarsely porous pellet. 

One can roughly estimate the effects of gas-phase diffusion at steady state using a simple ID diffusion model, which has been employed (in some form) by numerous workers. 343 This approach yields the following expression for the linearized steady-state chemical resistance due to binary diffusion of O2 in a stagnant film of thickness... 

If the melting is controlled by diffusion in the melt (which means extremely rapid interface reaction rate), the melt composition at the titanite-melt interface would be A, and that at the anorthite-melt interface would be B (Figure 4-35b). Treat the diffusion as binary diffusion. The diffusive flux across the melt at steady state is... 

Connection between Transport Processes and Solid Microstructure. The formation of cellular and dendritic patterns in the microstructure of binary crystals grown by directional solidification results from interactions of the temperature and concentration fields with the shape of the melt-crystal interface. Tiller et al. (21) first described the mechanism for constitutional supercooling or the microscale instability of a planar melt-crystal interface toward the formation of cells and dendrites. They described a simple system with a constant-temperature gradient G (in Kelvins per centimeter) and a melt that moves only to account for the solidification rate Vg. If the bulk composition of solute is c0 and the solidification is at steady state, then the exponential diffusion layer forms in front of the interface. The elevated concentration (assuming k < 1) in this layer corresponds to the melt that solidifies at a lower temperature, which is given by the phase diagram (Figure 5) as... 

For the solution of a salt composed of two ionizable species (binary electrolyte), the four basic equations can be combined to yield the convective diffusion equation for steady-state systems ... 

In this case, it is well known that the process occurs in steady state. To understand this process, one must consider it as a special case of binary diffusion, where the diffusivity of the Pd atoms is zero. Consequently, the frame of reference is the fixed coordinates of the solid Pd thin film. The interdiffusion or chemical diffusion coefficient is the diffusivity of the mobile species [20], that is, hydrogen. Then, the hydrogen flux in the Pd thin film is given by... 

Consider the problem of steady-state one-dimensional diffusion in a mixture of ideal gases. At constant T and P, the total molar density, c = P/RT is constant. Also, the Maxwell-Stefan diffusion coefficients D m reduce to binary molecular diffusion Dim, which can be estimated from the kinetic theory of gases. Since Dim is composition independent for ideal gas systems, Eq. (6.61) becomes... 

Law Simplified flux equations that arise from Eqs. (5-189) and (5-190) can be used for unidimensional, steady-state problems with binary mixtures. The boundary conditions represent the compositions xAl and xA(l at the left-hand and right-hand sides of a hypothetical layer having thickness Az. The principal restriction of the following equations is that the concentration and diffusivity are assumed to be constant. As written, the flux is positive from left to right, as depicted in Fig. 5-25. 

The experimentally performed transport processes used for evaluation of transport parameters include counter current binary or multicomponent gas diffusion under steady-state or chromatographic conditions, steady permeation of simple gases, dynamics of combined transport of binary or multicomponent gas mixtures, etc. Of significance, however, is that no automatic commercial instrument is available for these processes. Thus, the necessary apparatuses must be homemade. To obtain the transport parameters with acceptable confidence large numbers of experiments is required. It would be, therefore, of significant importance if at least part of the transport parameters could be obtained from standard textural analysis. 

The classical linear stability theory for a planar interface was formulated in 1964 by Mullins and Sekerka. The theory predicts, under what growth conditions a binary alloy solidifying unidirectionally at constant velocity may become morphologically unstable. Its basic result is a dispersion relation for those perturbation wave lengths that are able to grow, rendering a planar interface unstable. Two approximations of the theory are of practical relevance for the present work. In the thermal steady state, which is approached at large ratios of thermal to solutal diffusivity, and for concentrations close to the onset of instability the characteristic equation of the problem... 

This section contains a simple introduction to steady state and unsteady species mole (mass) diffusion in dilute binary mixtures. First, the physical interpretations of these diffusion problems are given. Secondly, the physical problem is expressed in mathematical terms relating the concentration profiles to the diffusion fluxes. Emphasis is placed on two diffusion problems that form the basis for the interfacial mass transfer modeling concepts used in reaction engineering. 

In the case of the fast binary reaction we could eliminate the reaction term from the reaction-diffusion-advection equation. But in general this is not possible. In this chapter we consider another class of chemical and biological activity for which some explicit analysis is still feasible. We consider the case in which the local-reaction dynamics has a unique stable steady state at every point in space. If this steady state concentration was the same everywhere, then it would be a trivial spatially uniform solution of the full reaction-diffusion-advection problem. However, when the local chemical equilibrium is not uniform in space, due to an imposed external inhomogeneity, the competition between the chemical and transport dynamics may lead to a complex spatial structure of the concentration field. As we will see in this chapter, for this class of chemical or biological systems the dominant processes that determine the main characteristics of the solutions are the advection and the reaction dynamics, while diffusion does not play a major role in the large Peclet number limit considered here. Thus diffusion can be neglected in a first approximation. 

Measurement of steady-state compositions y and y after thermal diffusion equilibrium has been established between temperatures T and T" is the most accurate way of determining the thermal diffusion constant. The next-to-the-last column of Table 14.26 pves values of the measured thermal diffusion constant for several binary isotopic mixtures. In all these cases, 7 is positive, which means that the light isotope concentrates at the higher temperature under the experimental conditions listed. 

In this section quantitative relationships for diffusion are discussed. Attention is focused on diffusion in a direction perpendicular to the interface between the phases and at a definite location in the equipment. Steady state is assumed, and the concentrations at any point do not change with time. This discussion is restricted to binary mixtures. 

A simple criterion whether the observed reaction rate in solid catalysts is not limited by mass transport has been derived by Weisz 61). If this criterion is not fulfilled, one can obtain the dimensiordess moduli 0 and 99 from the observed rate and known or estimated values of the respective diffusion coefficient, as described in the literature (57, 59, 60). In the case of a pelleted zeolite, these methods have first to be applied to the bulk pellet and subsequently to the individual crystals. The procedure requires that the concentration (ca)o at the phase boundary of the zeolite crystals and the binary diffusivity at steady state are known must be estimated from transient measurements in the same... 

In the discussion so far, the diffusional and electrical fluxes of the ionic and electronic carriers were treated separately. However, as will become amply clear in this section and was briefly touched upon in Sec. 5.6, in the absence of an external circuit such as the one shown in Fig. 7.7, the diffusion of a charged species by itself is very rapidly halted by the electric field it creates and thus cannot lead to steady-state conditions. For steady state, the fluxes of the diffusing species have to be coupled such that electroneutrality is maintained. Hence, in most situations of great practical importance such as creep, sintering, oxidation of metals, efficiency of fuel cells, and solid-state sensors, to name a few, it is the coupled diffusion, or ambipolar diffusion, of two fluxes that is critical. To illustrate, four phenomena that are reasonably well understood and that are related to this coupled diffusion are discussed in some detail in the next subsections. The first deals with the oxidation of metals, the second with ambipolar diffusion in general in a binary oxide, the third with the interdiffusion of two ionic compounds to form a solid solution. The last subsection explores the conditions for which a solid can be used as a potentiometric sensor. 

Consider steady-state diffusion only in the z direction without chemical reaction in a binary gaseous mixture. For the case of one-dimensional diffusion, equation (1-20) becomes... 

Consider the following numerical example. A binary gaseous mixture of components A and B at a pressure of 1 bar and temperature of 300 K undergoes steady-state equimolar counterdiffusion along a 1-mm-thick diffusion path. At one end of the path... 

The reaction is very rapid, so that the partial pressure of CO at the metal surface is essentially zero. The gases diffuse through a film 0.625 mm thick. At steady-state, estimate the rate of production of nickel carbonyl, in mole/m2 of solid surface per second. The composition of the bulk gas phase is 50 mol% CO. The binary gas diffusiv-ity under these conditions is DAB = 20.0 mm2/s. 

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