Diffusion coefficient variable with time

When the diffusion coefficient varies with time, the fundamental transformation commonly used consists in defining a new time variable t as... 

Chloride ion diffusion in a one-dimensional slab has been modeled by Lin [7] subject to time-dependent surface concentration and diffusion coefficient. Bazant [68] developed a physical model to study reinforcement corrosion. Saetta et al. [69] considered ion diffusion coefficient variability with concrete parameters in a chloride diffusion study in partially saturated concrete. The time for corrosion initiation has been empirically... 

There are further classes of experiment that result in pseudo-2D NMR spectra. These do not have a second frequency axis resulting from Fourier transformation of a variable time, but the second axis is some other parameter. One example is provided by continuous-flow directly coupled HPLC-NMR spectra where the second axis in the pseudo-2D plot is the chromatographic retention time. Another example is diffusion-ordered NMR spectroscopy where the second axis plots the molecular diffusion coefficient associated with each NMR peak, this parameter being derived from the dependence of peak intensities on the square of an applied magnetic field gradient. 

We have already met the concept of error propagation a few times when dealing with the change of variable formulas for probability distribution, but let us try to illustrate it with a simple example. We want to measure the diffusion coefficient Q) of uranium in a glass by maintaining at a specific temperature and for a specific time t the surface of one long glass rod in contact with a concentrated solution of uranium. We admit without further justification (see Section 8.5) that the depth x of uranium... 

H is the plate height (cm) u is linear velocity (cm/s) dp is particle diameter, and >ni is the diffusion coefficient of analyte (cm /s). By combining the relationships between retention time, U, and retention factor, k tt = to(l + k), the definition of dead time, to, to = L u where L is the length of the column, and H = LIN where N is chromatographic efficiency with Equations 9.2 and 9.3, a relationship (Equation 9.4) for retention time, tt, in terms of diffusion coefficient, efficiency, particle size, and reduced variables (h and v) and retention factor results. Equation 9.4 illustrates that mobile phases with large diffusion coefficients are preferred if short retention times are desired. 

Although MC does not have a proper time variable, the time equivalent of MC moves can be estimated, for example, by computing rotational correlation functions or molecular diffusion coefficients in well characterized liquids these are time-dependent quantities also experimentally accessible, and the comparison between the number of MC moves and the corresponding experimental data provides the required time equivalents. An order of magnitude estimate with typical MC translational or rotational steps of 0.2 A and 4° gives 1 or 2 ps per 10 translational or rotational MC moves, respectively. 

Because the fluid is in equilibrium, any ensemble average property should not change with time. Hence, the ensemble average of (u(tf)u(t")> depends only on the relative difference of time, t — t". That is, it is a stationary process. On transforming the time variables to f and r = tr — f" (rather like the centre of diffusion coefficient transformation of Chap. 9, Sect. 2), the Green—Kubo expression for the diffusion coefficient is obtained [453, 490],... 

Equation (2.11) refers to the flux conservation and Eq. (2.12) to the establishment of the nemstian equilibrium. Note that under these conditions, the original problem in terms of variables x and t has been transformed into a one-variable problem (s ), that is, c0 and cR can be expressed only as functions of the variables o and sR, respectively (which include distance and time variables), because they diffuse with different diffusion coefficients D0 and DR. This problem can now be solved by making = dci/dsf, and Eq. (2.9) becomes... 

The general solutions of the fundamental systems of nonlinear equations [Eq. (2)] will be of the type wherein the state variables are dependent both on time and space, which will manifest in the form of wave propagation. Coupling between several parts of the system will be transmitted through the generalized diffusion coefficient D. If the associated transport process proceeds on a time scale comparable to or slower than the period of the temporal oscillation, macroscopic wave propagation phenomena are to be expected, as, for example, realized with the Belousov-Zhabotinsky... 

Buckinghams 7r-theorem [i] predicts the number of -> dimensionless parameters that are required to characterize a given physical system. A relationship between m different physical parameters (e.g., flux, - diffusion coefficient, time, concentration) can be expressed in terms of m-n dimensionless parameters (which Buckingham dubbed n groups ), where n is the total number of fundamental units (such as m, s, mol) required to express the variables. For an electrochemical system with semiinfinite linear geometry involving a diffusion coefficient (D, units cm2 s 1), flux at x = 0 (fx=o> units moles cm-2 s 1), bulk concentration (coo> units moles cm-3) and time (f, units s), m = 4 (D, fx=0, c, t) and n - 3 (cm, s, moles). Thus m-n - 1 therefore only one dimensionless parameter can be constructed and that is fx=o (t/Dy /coo. Dimensional analysis is a powerful tool for characterizing the behavior of complex physical systems and in many cases can define relationships... 

Dimensionless analysis — Use of dimensionless parameters (-> dimensionless parameters) to characterize the behavior of a system (- Buckinghams n-theorem and dimensional analysis). For example, the chronoampero-metric experiment (-> chronoamperometry) with semiinfinite linear geometry relates flux at x = 0 (fx=o, units moles cm-2 s-1), time (t, units s-1), diffusion coefficient (D, units cm2 s-1), and concentration at x = oo (coo, units moles cm-3). Only one dimensionless parameter can be created from these variables (-> Buckingham s n-theorem and dimensional analysis) and that is fx=o (t/D)1/2/c0C thereby predicting that fx=ot1 2 will be a constant proportional to D1/,2c0O) a conclusion reached without any additional mathematical analysis. Determining that the numerical value of fx=o (f/D) 2/coo is 1/7T1/2 or the concentration profile as a function of x and t does require mathematical analysis [i]. 

When ux = uy = uz = 0, indicating no convective motion of the gas, Eq. 10.15 reverts to the pure diffusion case. The terms ux, uy, and uz are not necessarily equal, nor are they usually constant, since convective velocities decrease as a surface is approached. Equation 10.15 thus represents a second-order partial differential equation with variable coefficients. These types of equations are usually quite difficult to solve. However, often it is sufficient to consider only the steady-state solution, i.e., the case where dc/dt = 0, indicating that the concentration at any point within the system is not changing with time. Then Eq. 10.15 becomes... 

In the usual macroscopic analysis of transfer phenomena, fluids are considered as continuous media and macroscopic properties are assumed to vary continuously in time and space. The physical properties (density,. ..) and macroscopic variables (velocity, temperature,...) are averages on a sufficient number of atoms or molecules. If A 10" is a number of molecules high enough to be significant, the side length of a volume containing these N molecules is about 70 nm for a gas in standard conditions and 8 nm for a liquid. These dimensions are smallest than those of a microchannel whose characteristic dimension is between 1 to 300 pm. The transport properties (heat and mass diffusion coefficients, viscosity) depend on the molecular interactions whose effects are of the order of magnitude of the mean free path These last effects can be appreciated with the Knudsen number... 

Transient heat conduction or mass transfer in solids with constant physical properties (diffusion coefficient, thermal diffusivity, thermal conductivity, etc.) is usually represented by a linear parabolic partial differential equation. In this chapter, we describe how one can arrive at the semianalytical solutions (solutions are analytical in the time variable and numerical in the spatial dimension) for linear parabolic partial differential equations using Maple, the method of lines and the matrix exponential. 

Barrow [772] derived a kinetic model for sorption of ions on soils. This model considers two steps adsorption on heterogeneous surface and diffusive penetration. Eight parameters were used to model sorption kinetics at constant temperature and another parameter (activation energy of diffusion) was necessary to model kinetics at variable T. Normal distribution of initial surface potential was used with its mean value and standard deviation as adjustable parameters. This surface potential was assumed to decrease linearly with the amount adsorbed and amount transferred to the interior (diffusion), and the proportionality factors were two other adjustable parameters. The other model parameters were sorption capacity, binding constant and one rate constant of reaction representing the adsorption, and diffusion coefficient of the adsorbate in tire solid. The results used to test the model cover a broad range of T (3-80°C) and reaction times (1-75 days with uptake steadily increasing). The pH was not recorded or controlled. 

As shown in later sections, the coefficient of diffusion is a function of particle size, with small particles diffusing more rapidly than larger ones. For a polydisperse aerosol, the concentration variable can be set equal to nj( ipr r, t)d(Jp) or h(iv, r, /) dv, and both sides of (2.3) can be divided by d(dp) or dv. Hence the equation of diffusion describes the changes in the particle size distnbntion with time and position as a result of diffusive processes. Solutions to the diffusion equation for many different boundary conditions in the absence of flow have been collected by Carslaw and Jaeger ( 959) and Crank (1975). 

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