Constant diffusion coefficient

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. 

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-... 

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 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. 

Another m3d h arises from the intuition that pressure effect is opposite to the temperature effect. This is not true in kinetics. Therefore, kinetic constants (reaction rate constants, diffusion coefficients, etc.) almost always increase with increasing temperature, but they may decrease or increase with increasing pressure. Both positive and negative pressure dependences are well accounted for by the transition-state theory and are not strange. 

The one-dimensional binary diffusion equation with constant diffusion coefficient is (Equation 3-10)... 

The isotopic fraction profiles may be described by a roughly constant diffusion coefficient across major concentration gradients. 

Henry s law constants, diffusion coefficients, and rate constants are known, extrapolation to the atmosphere can be carried out reliably and reasonably accurately. 

Obviously, in the case of isotopic exchange, a = 1. Furthermore, in isotopic exchange, the diffusion coefficients are equal to the self-diffusion coefficient of each ion. For different ions (not isotopes), the self-diffusion coefficient is substituted by an empirical constant diffusion coefficient. In the general case of ion exchange, the diffusion coefficient is not constant and for practical reasons, an appropriate average value is used. For the case of complete conversion into A-form (infinite solution volume), this average diffusion coefficient is (Helfferich, 1962)... 

The behaviour we are expecting to emerge from this physico-chemical model is that of a steady wave of reaction moving from left to right in Fig. 11.2 into the region of unreacted A. By steady, we really mean that the wavefront should maintain its shape as it moves with a constant speed. It is this shape and speed which we seek to determine (and express in terms of the rate constant, diffusion coefficient, etc.). 

Returning to the original rate constants, diffusion coefficient, etc., this gives the speed as... 

So far, it appears that the gas transport properties of glassy polymer membranes, manifested in a decreasing P(a), or increasing D(C), function can be adequately represented by the above dual diffusion model with constant diffusion coefficients Dl5 D2 (or Dtj, DX2). We now consider the implications of this model from the physical point of view ... 

Assuming that the constant diffusion coefficients are equal to Z)avg, and the concentration dependent parameters fi are as above, modified diffusion coefficients were calculated from ... 

As mentioned, all reaction models will include initially unknown reaction parameters such as reaction orders, rate constants, activation energies, phase change rate constants, diffusion coefficients and reaction enthalpies. Unfortunately, it is a fact that there is hardly any knowledge about these kinetic and thermodynamic parameters for a large majority of reactions in the production of fine chemicals and pharmaceuticals this impedes the use of model-based optimisation tools for individual reaction steps, so the identification of optimal and safe reaction conditions, for example, can be difficult. 

In a spatially extended system, fluctuations that are always present cause the variables to differ somewhat in space, inducing transport processes, the most common one being diffusion. In the case of constant diffusion coefficients /), the system s dynamics is then governed by reaction-diffusion equations ... 

Fig.12 Comparison of the measured course of drying (— —) with the calculated one assuming constant diffusion coefficients (108 D = 0.5-10) from [28]...

Despite the large number of analytical solutions available for the diffusion equation, their usefulness is restricted to simple geometries and constant diffusion coefficients. However, there are many cases of practical interest where the simplifying assumptions introduced when deriving analytical solutions are unacceptable. Such a case, for example, is the diffusion in polymer systems characterized by concentration-dependent diffusion coefficients.This chapter gives an overview of the most powerful numerical methods used at present for solutions of the diffusion equation. Indeed the application of these methods in practice needs the use of adequate computer programs (software). 

Eqs. (7-5) and (7-6) are known as Fick s second law for the case where the diffusion has a constant diffusion coefficient. 

In addition to analytical solutions the possibility exists to obtain numerous exact solutions using numerical methods with help from computers. The advantage of numerical methods lies primarily with their application for complicated cases, e.g. for non-constant diffusion coefficients, for which there are no analytical solutions. 

We consider for now the one-dimensional diffusion equation, with constant diffusion coefficient D ... 

Assuming constant diffusion coefficient, the equation describing the radial diffusion in cylindrical coordinates may be written ... 

The radial diffusion equation in spherical coordinates may be written for constant diffusion coefficient as ... 

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