Constant Diffusivity

When D is constant, Eq. 4.2 takes the relatively simple form of the linear second-order partial differential equation 

Some of the major features of this equation are discussed below, and methods of solving it under a variety of boundary and initial conditions are described at length in Chapter 5. 

1 Geometrical Interpretation of the Diffusion Equation when Diffusivity is Constant 

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An alternative method known as slicing and scaling has been developed (23,24). In this, the rate of diffusion is determined on a thin specimen (6—10 mm thick) and a scaling factor S used to relate the results to a thick specimen. For a material satisfying the requirements of a constant diffusion and constant initial pressure,, the same ratio of time thickness provides the same values of p and %. Thus the thermal resistance of a specimen of thickness at time can be obtained by conditioning a specimen of thickness over a time given by... 

Eig. 7. Drying of com kernels by Hquid diffusion. The dashed line is that predicted by theory based on constant diffusivity. The solid curve shows actual... 

Diffusion equations mav also be used to study vapor diffusion in porous materi s. It should be clear that aU estimates based on relationships that assume constant diffusivity are approximations. Liquid diffusivity in sohds usually decreases with moisture concentration. Liquid and vapor diffusivity also change, and material shrinks during diying. 

In case of a steady state with a constant diffusion layer thickness for all ions it follows [for simplicity the surfix (aq) is ommitted] ... 

In conclusion, therefore, the model proposed by Gal-Or and Resnick predicts mass-transfer rates that correlate reasonably well with the experimental data now available. Further experimental work as well as accurate values for the reaction-rate constants, diffusivities, and solubilities for other... 

Equation (8.12) is a form of the convective dijfusion equation. More general forms can be found in any good textbook on transport phenomena, but Equation (8.12) is sufficient for many practical situations. It assumes constant diffusivity and constant density. It is written in cylindrical coordinates since we are primarily concerned with reactors that have circular cross sections, but Section 8.4 gives a rectangular-coordinate version applicable to flow between flat plates. 

The diffusion system. Figure 8.31(B), is a useful and simple apparatus for preparing mixtures of volatile and moderately volatile vapors in a gas stream [388]. The method is based on the constant diffusion of a vapor from a tube of accurately known dimensions, producing a gas phase concentration described by equation (8.12). 

Consider a long circular cylinder in which a solute diffuses radially. The concentration is a function of radial position (r) and time (t). In the case of constant diffusion coefficient, the diffusion equation is... 

A sphere is assumed to be a poorly soluble solute particle and therefore to have a constant radius rQ. However, the solid solute quickly dissolves, so the concentration on the surface of the sphere is equal to its solubility. Also, we assume we have a large volume of dissolution medium so that the bulk concentration is very low compared to the solubility (sink condition). The diffusion equation for a constant diffusion coefficient in a spherical coordinate system is... 

To calculate the mean escape time over a potential barrier, let us apply the Fokker-Planck equation, which, for a constant diffusion coefficient D = 2kT/h, may be also presented in the form... 

In order to achieve the most simple presentation of the calculations, we shall restrict ourselves to a one-dimensional state space in the case of constant diffusion coefficient D = 2kT/h and consider the MFPT (the extension of the method to a multidimensional state space is given in the Appendix of Ref. 41). Thus the underlying probability density diffusion equation is again the Fokker-Planck equation (2.6) that for the case of constant diffusion coefficient we present in the form ... 

In this section we will consider this approach in detail for different types of potential profiles (p(x) = x)/k,T, and to avoid cumbersome calculations we present the analysis for the constant diffusion coefficient D = 2kT/h, but the results, of course, may be easily generalized for any D(x) / 0. 

One can check that this result coincides with the result by Risken and Jung (5.24) for the case of constant diffusion coefficient. 

The rate equation is given by item A in Table 16-11. With pore fluid and adsorbent at equilibrium at each point within the particle and for a constant diffusivity, the rate equation can be written as ... 

That is, for conditions where the diffusion distance is kept constant at various temperatures, D will have a weak temperature dependence, and Dh will be thermally activated. For constant diffusion distance, it is the time to diffuse a distance L that is thermally activated. 

The lower activation energies found for p-type a-Si H using the evolution technique may be due to microstructure in the films grown with a He carrier gas. The higher D0 obtained in doped samples using the concentration profiling technique results from the correction of the data to a constant diffusion distance L = 1000 A (Jackson et al., 1989a). 

Ccs is the constant concentration of the diffusing species at the surface, c0 is the uniform concentration of the diffusing species already present in the solid before the experiment and cx is the concentration of the diffusing species at a position x from the surface after time t has elapsed and D is the (constant) diffusion coefficient of the diffusing species. The function erf [x/2(Dr)1/2] is called the error function. The error function is closely related to the area under the normal distribution curve and differs from it only by scaling. It can be expressed as an integral or by the infinite series erf(x) — 2/y/ [x — yry + ]. The comp-... 

Fig. 18b.3. Ideal concentration gradients of O and R during a redox reaction on the surface of the working electrode. The x-axis shows the distance from the electrode surface. Bulk solution is stirred to maintain a constant diffusion layer thickness, d.

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... 

Let us assume parallel flux in a semi-infinite medium bound by the plane x=0. Diffusion of a given element takes place from the plane x=0 kept at concentration Cint. Introducing a Boltzmann variable u with constant diffusion coefficient such as... 

When combined with the Fourier expansion of functions, separation of variables is another powerful method of solutions which is particularly useful for systems of finite dimensions. Regardless of boundary conditions, we decompose the solution C(x, t), where the dependence of C on x and t is temporarily emphasized, to the general one-dimensional diffusion equation with constant diffusion coefficient... 

In the diffusion equation (8.6.3) with radial flux and constant diffusion coefficient, let us introduce the new variable u r,t) = u=Cr... 

Let us assume that a sphere with radius a is immersed in a liquid of finite volume, e.g., a mineral in a hydrothermal fluid. Diffusion in liquids is normally fast compared to diffusion in solids, so that the liquid can be thought of as homogeneous. Similar conditions would apply to a sphere degassing into a finite enclosure, e.g., for radiogenic argon loss in a closed pore space. Given the diffusion equation with radial flux and constant diffusion coefficient... 

All the solutions with constant diffusion coefficients can therefore be used for problems with time-dependent diffusion coefficients upon replacement of 3>t/X2 by t. 

Rafler et al. showed in an early work [102] that the diffusion coefficient of EG varies with the overall effective polycondensation rate and they proposed a dependency of the diffusion coefficient on the degree of polycondensation. This dependency is obvious, because the diffusion coefficient is proportional to the reciprocal of the viscosity which increases by four orders of magnitude during polycondensation from approximately 0.001 Pas (for Pn = 3) to 67Pas (for Pn = 100) at 280 °C. In later work, Rafler et al. [103, 104, 106] abandoned the varying diffusion coefficient and instead added a convective mass-transport term to the material balance of EG and water. The additional model parameter for convection in the polymer melt and the constant diffusion coefficient were evaluated by data fitting. 

The second approach employs a detailed reaction model as well as the diffusion of EG in solid PET [98, 121-123], Commonly, a Fick diffusion concept is used, equivalent to the description of diffusion in the melt-phase polycondensation. Constant diffusion coefficients lying in the order of Deg, pet (220 °C) = 2-4 x 10 10 m2/s are used, as well as temperature-dependent diffusion coefficients, with an activation energy for the diffusion of approximately 124kJ/mol. 

In a theoretical study, Chorny and Krasuk(4) analysed the diffusion process in extraction from simple regular solids, assuming constant diffusivity. 

These are contradictory modifications of the basic assumption the constant diffusivities are now functions of the distance from the emission source and of the stability of the atmosphere. The success in the application of the Gaussian plume model may be justification enough. 

In another review, Hoffert discussed the social motivations for modeling air quality for predictive purposes and elucidated the components of a model. Meteorologic factors were summarized in terms of windfields and atmospheric stability as they are traditionally represented mathematically. The species-balance equation was discussed, and several solutions of the equation for constant-diffusion coefficient and concentrated sources were suggested. Gaussian plume and puff results were related to the problems of developing multiple-source urban-dispersion models. Numerical solutions and box models were then considered. The review concluded with a brief outline of the atmospheric chemical effects that influence the concentration of pollutants by transformation. 

Conditions of constant potential are frequently employed in laboratory scale experiments. In these experiments, the cunent tlirough the cell falls with time due to depletion of the substrate. Under conditions of constant diffusion layer thickness, the current it at time t is given by Equation 1.4 [17] where D is the diffusion coeffi-... 

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