One-dimensional diffusion theory

The exhalation of Rn from material surfaces is controlled by the generation rate of Rn in the material, and the transport by diffusion through the material to the surface. The generation rate is determined by the Rn content of the material, and the emanation fraction. The transport through the material is controlled by the diffusion length through the material. The diffusion process is well described mathematically by one-dimensional diffusion theory, so that knowledge of these parameters will allow accurate calculation of the Rn exhalation rate from the material surface. 

Assuming the one-dimensional diffusion theory is applicable, the released mass per unit area in one direction up to time t , M(t) can be written in the following form, if the solute surface concentration is constant [20]. 

H. P. Flatt, (AlM-6) A Multi-Group, One Dimensional Diffusion Theory Code, Unpublished Atomics Ihtema-tional Report (1960). 

A one-dimensional diffusion theory code developed by General Atomic (report unpublished). 

D. A. MENELEY, L. C. KVITEK, and D. M. O SHEA, MACH 1, A One-Dimensional Diffusion Theory Package," ANL-7223, Argonne National Laboratory (June 1966). 

Few data cvirrently exist on the ellectlveness of boron and cadmium for criticality prevention In operations external to reactors that may involve fuel elements under conditions of water Immersion. Material bucklings and extrapolation distances have previously been measured and reported for 25.2 wt% Pu02-U(Nat)0a fuel pins in water. These experimental results have also teen compared with one-dimensional diffusion theory calculations using ENDF/B version n cross-section data. The previous measurements have now been extended to include criticality data On these same fuel pins positioned in lattices with boron- and cadmium-poisoned water. 

H. P. FLATT and D. C BALLER, AIM-5-A Multi-group One Dimensional Diffusion Theory Code, NAA-SR-4694. 

The infinite medium with one-dimensional diffusion and constant diffusion coefficient can be treated easily with the point source theory. Let us first assume that two half-spaces with uniform initial concentrations C0 for x < 0 and 0 for x > 0 are brought into contact with each other. The amount of substance distributed per unit surface between x and x + dx is just C0dx. From the previous result, at time t the effect of the point source C0 dx located at x on the concentration at x will be... 

A model of such structures has been proposed that captures transport phenomena of both substrates and redox cosubstrate species within a composite biocatalytic electrode.The model is based on macrohomo-geneous and thin-film theories for porous electrodes and accounts for Michaelis—Menton enzyme kinetics and one-dimensional diffusion of multiple species through a porous structure defined as a mesh of tubular fibers. In addition to the solid and aqueous phases, the model also allows for the presence of a gas phase (of uniformly contiguous morphology), as shown in Figure 11, allowing the treatment of high-rate gas-phase reactant transport into the electrode. 

The second part (sections H and I) is devoted to a detailed discussion of the dynamics of unimolecular reactions in the presence and the absence of a potential barrier. Section H presents a critical examination of the Kramers approach. It is stressed that the expressions of the reaction rates in the low-, intermediate-, and high-friction limits are subjected to restrictive conditions, namely, the high barrier case and the quasi-stationary regime. The dynamics related to one-dimensional diffusion in a bistable potential is analyzed, and the exactness of the time dependence of the reaction rate is emphasized. The essential results of the non-Markovian theory extending the Kramers conclusions are also discussed. The final section investigates in detail the time evolution of an unimolecular reaction in the absence of a potential barrier. The formal treatment makes evident a two-time-scale description of the dynamics. 

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... 

As our first application of the linearized theory we consider steady-state, one-dimensional diffusion. This is the simplest possible diffusion problem and has applications in the measurement of diffusion coefficients as discussed in Section 5.4. Steady-state diffusion also is the basis of the film model of mass transfer, which we shall discuss at considerable length in Chapter 8. We will assume here that there is no net flux = 0. In the absence of any total flux, the diffusion fluxes and the molar fluxes are equal = J. ... 

In 1964 Toor and Stewart and Prober independently put forward a general approach to the solution of multicomponent diffusion problems. Their method, which was discussed in detail in Chapter 5, relies on the assumption of constancy of the Fick matrix [D] along the diffusion path. The so-called Tinearized theory of Toor, Stewart, and Prober is not limited to describing steady-state, one-dimensional diffusion in ideal gas mixtures (as we have already demonstrated in Chapter 5) however, for this particular situation Eq. 5.3.5, with [P] given by Eq. 4.2.2, simplifies to... 

Now, it is necessary to discuss the mass transfer coefficient for component j in the boundary layer on the vapor side of the gas-liquid interface, fc ,gas, with units of mol/(area-time). The final expression for gas is based on results from the steady-state film theory of interphase mass transfer across a flat interface. The only mass transfer mechanism accounted for in this extremely simple derivation is one-dimensional diffusion perpendicular to the gas-liquid interface. There is essentially no chemical reaction in the gas-phase boundary layer, and convection normal to the interface is neglected. This problem corresponds to a Sherwood number (i.e., Sh) of 1 or 2, depending on characteristic length scale that is used to define Sh. Remember that the Sherwood number is a dimensionless mass transfer coefficient for interphase transport. In other words, Sh is a ratio of the actual mass transfer coefficient divided by the simplest mass transfer coefficient when the only important mass transfer mechanism is one-dimensional diffusion normal to the interface. For each component j in the gas mixture. 

Further relaxation of the assumptions of the film and penetration theories was suggested by Danckweits who viewed the process as one of transient one-dimensional diffusion to packets or elements of fluid that reside at the phaM interface for varying periods of time. Therefore, the model is that of the penetration theory with a distribution of contact times. The surface age distribution (r) is defined such that is the fraction of surface that has resided at the interface for a time between I and i + dt. The mass transfer flux for the entire surface is obtained by integration of the instantaneous flux over all exposure times ... 

To start, we consider one-dimensional diffusion model. This is not only because water diffusion into polymer matrix is the first step but the mathematics of two- and three-dimensional diffusion is more complicated. In fact, the results obtained from one-dimensional diffusion model has been practically used whenever diffusion kinetics under investigation. The kinetic models obtained from one-dimensional theory are enough to cover the kinetic models often used by most researchers in the world. Two- or three-dimensional diffusion follows the same principle. 

The resulting homogenized cross sections were then used in Convendonal one- and two-dimensional diffusion theory calcuiadons on the as-budt core dimensions to obtain the appropriate keff s and flux distributions. Onedimensional and reduced-groiqji two-dimensional transport dieory calcuhitlons were performed to obtain extrapolated S eorrecdon factors. 

Calculational methods used for criticality safety evaluations are commonly one-dimensional (diffusion or tran rt theory) or three-dimensional Monte Carlo. Suitable bench-maik experiments for the former ate also necessarily essentially one-dimensional (Le., experiments with spheres or with series of cylinders or cuboids that permit extrapolation to infinite height or extent). In principle, any experimental configuration may be represented by the Monte Carlo geometry package, but if the configuration is complex, it will be... 

The limits theory for a one-dimensional diffusion flame was developed by Zeldovich Ya. B. [6] and Spalding D. [7]. Investigation of laminar diffusion flames was started in [6-9]. A comprehensive representation of the theoretical methods can be found in [10]. 

Keilson-Storer kernel 17-19 Fourier transform 18 Gaussian distribution 18 impact theory 102. /-diffusion model 199 non-adiabatic relaxation 19-23 parameter T 22, 48 Q-branch band shape 116-22 Keilson-Storer model definition of kernel 201 general kinetic equation 118 one-dimensional 15 weak collision limit 108 kinetic equations 128 appendix 273-4 Markovian simplification 96 Kubo, spectral narrowing 152... 

Surface Renewal Theory. The film model for interphase mass transfer envisions a stagnant film of liquid adjacent to the interface. A similar film may also exist on the gas side. These h5q>othetical films act like membranes and cause diffu-sional resistances to mass transfer. The concentration on the gas side of the liquid film is a that on the bulk liquid side is af, and concentrations within the film are governed by one-dimensional, steady-state diffusion ... 

In order to illustrate the effects of media structure on diffusive transport, several simple cases will be given here. These cases are also of interest for comparison to the more complex theories developed more recently and will help in illustrating the effects of media on electrophoresis. Consider the media shown in Figure 18, where a two-phase system contains uniform pores imbedded in a matrix of nonporous material. Solution of the one-dimensional point species continuity equation for transport in the pore, i.e., a phase, for the case where the external boundaries are at fixed concentration, Ci and Cn, gives an expression for total average flux... 

For the radical neutrals, boundary conditions are derived from diffusion theory [237, 238]. One-dimensional particle diffusion is considered in gas close to the surface at which radicals react (Figure 14). The particle fluxes in the two z-directions can be written as... 

The UMEs used in bioarrays can be divided into three types disk, ring, and strip electrodes. The theory of the disk, ring, and strip UMEs has been extensively studied [97-100], Due to the edge effect, the profile of the mass diffusion to the ultramicroelectrode surface is three dimensional, and can significantly enhance the mass transportation in comparison to the conventional large electrode with one-dimensional mass transportation. The steady-state measurement at a planar UME can be expressed as... 

The thermal healing has been studied most extensively for one-dimensional gratings. Above roughening, the gratings acquire, for small amplitude to wavelength ratios, a sinusoidal form, as predicted by the classical continuum theory of Mullins and confirmed by experimenf-s and Monte Carlo simulations. - The decay of the amplitude is, asymptotically, exponential in time. This is true for both evaporation dynamics and (experimentally more relevant) surface diffusion. 

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