Diffusivity as a Function of Time

Equation 4.51 is an integral equation that can be used to determine D(c ) by a graphical construction or numerical solution. The derivative required in Eq. 4.51 is provided by the measured concentration profile at time t and the integration is performed on the inverse of c x) [6]. However, this historically important method is only moderately accurate, and it would be preferable to obtain diffusion profiles for various assumed diffusivities as a function of concentration by computation. D(c) could be deduced by fitting calculated results for a parametric representation of D(c) to an experimentally determined diffusion profile. 

When D is a function of time, but not position, Eq. 4.2 takes the form 

This could be the case for a diffusion specimen that is slowly cooled while a uniform temperature is maintained. Problems of this type can be treated by making the change of variable 

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Figure 2.64 Schematic diagram of diffusion as a function of time.

In this paper, the spin exchange experiments, which employ the Dante sequence, are used to probe the resonance profile of those carbons which lie in an 0.7-1.0 nm sphere surrounding the resonance perturbed by the Dante sequence. A detailed description of the method and a consideration of the spatial extent of spin exchange (or spin diffusion) as a function of time are beyond the scope of this paper. Such information may be found in a forthcoming paper (32). In the present paper the Dante sequence will be viewed as a method for very selectively perturbing the Zeeman spin population within a multiplet. spin exchange in cellulose will be looked upon... 

Figure 4b. Graph of the normalized concentration due to cylindrical diffusion as a function of time, t,for D=1 dO mfs and Rq=150 jum.

Integration of Eq. (3.7.1) presents some difficulty since our present knowledge of sintering kinetics is insufficient to allow the prediction of the effective diffusivity as a function of time and position in the ash layer. 

LID) see Ref. 139. In this last method, a small area, about 0.03 cm radius, is depleted by a laser beam, and the number of adatoms, N(t), that have diffused back is found as a function of time. From Pick s second law of diffusion ... 

An important advancement in carburizing has been the development of diffusion models to calculate the carbon gradient as a function of time as the gas composition and temperature change (13). Such models can be coupled with computer control of the gas composition and temperature to produce desired carbon profiles. 

Siemes and Weiss (SI4) investigated axial mixing of the liquid phase in a two-phase bubble-column with no net liquid flow. Column diameter was 42 mm and the height of the liquid layer 1400 mm at zero gas flow. Water and air were the fluid media. The experiments were carried out by the injection of a pulse of electrolyte solution at one position in the bed and measurement of the concentration as a function of time at another position. The mixing phenomenon was treated mathematically as a diffusion process. Diffusion coefficients increased markedly with increasing gas velocity, from about 2 cm2/sec at a superficial gas velocity of 1 cm/sec to from 30 to 70 cm2/sec at a velocity of 7 cm/sec. The diffusion coefficient also varied with bubble size, and thus, because of coalescence, with distance from the gas distributor. 

Fluctuation-diffusion current, 284 in corrosion processes, 302 as a function of nickel ion concentration, 289 as a function of time, 288 Fluorine... 

These results have been fit to experimental data obtained for the reaction between a diisocyanate and a trifunctional polyester polyol, catalyzed by dibutyltindilaurate, in our laboratory RIM machine (Figure 2). No phase separation occurs during this reaction. Reaction order, n, activation energy, Ea, and the preexponential factor. A, were taken as adjustable parameters to fit adiabatic temperature rise data. Typical comparison between the experimental and numerical results are shown in Figure 7. The fit is quite satisfactory and gives reasonable values for the fit parameters. Figure 8 shows how fractional conversion of diisocyanate is predicted to vary as a function of time at the centerline and at the mold wall (remember that molecular diffusion has been assumed to be negligible). 

It is possible to model what a single molecule of water is doing in the presence of bulk water. It is necessary to distinguish one molecule as being different, say a different color and identity within the program. The state and the movement rules are kept the same for this molecule as for the rest of the water molecules. The movement of the designated water molecule away from a reference point as a function of time is the diffusion rate. 

Spontaneous dewetting takes place by the formation of a droplet connected to the second layer. As a function of time, the droplets increase in volume while the area covered by the second layer decreases. In cases where the bare substrate was partly exposed, the diffusion and aggregation of second-layer molecules into droplets preserved the exposed regions of the substrate, as shown in Figure 20. 

Sequence of Phases Formed as a Function of Time, As Caused by the Prevailing Diffusion Reactions... 

Chronopotentiometry has been widely used to determine diffusion coefficients in molten salts. Chronopotentiometry is an experimental procedure in which the potential of an electrode is observed as a function of time during the passage of a constant current sufficiently large to produce concentration polarization with respect to the species undergoing electrochemical reaction. 

Figure 34. Plots of x as a function of time for a typical experiment in heat-transfer salts at 168°C. (Reprinted from 0. Odawara, 1. Okada, and K. Kawamura, "Measurement of Thermal Diffusivity of HTS by Optical Interferometry," J. Chem. Eng. Data 22 222-225, Copyright 1977 American Chemical Society.)...

It has been known for 40 years, since the initial PGSE experiment of Stejskal and Tanner [20], that NMR is an excellent tool for measuring the statistical displacements of molecules with time, usually called diffusion. The exponential attenuation a of the PGSE signal as a function of time between the PGSE pulses is given by [21]... 

The reaction is carried out in close-loop reactor connected to a mass spectrometer for 1S02, 180160 and 1602 analyses as a function of time [38], The gases should be in equilibrium with the metallic surface (fast adsorption/desorption steps 1 and f ) If the bulk diffusion is slow (step 6) and the direct exchange (step 5) does occur at a negligible rate, coefficients of surface diffusion Ds can be calculated from the simple relationship between the number of exchanged atoms Ne and given by the model of circular sources developed by Kramer and Andre [41] ... 

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