The diffusion coefficient varies with time

Using equation (8.6.6), we obtain the mean concentration of the sphere as 

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

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. 

The fraction of total 37Ar degassed after the /cth heating step is 

---

In both cases, the critical transport parameter is the chloride diffusion coefficient. The diffusion coefficient varies with water to cement ratio and time. In this model, the effective chloride diffusion coefficient is derived from concrete permeabihty, water/ cement ratio, and concrete resistivity. 

In a solution of molecules of uniform molecular weight, all particles settle with the same value of v. If diffusion is ignored, a sharp boundary forms between the top portion of the cell, which has been swept free of solute, and the bottom, which still contains solute. Figure 9.13a shows schematically how the concentration profile varies with time under these conditions. It is apparent that the Schlieren optical system described in the last section is ideally suited for measuring the displacement of this boundary with time. Since the velocity of the boundary and that of the particles are the same, the sedimentation coefficient is readily measured. 

Particle-Diffusion Control. Here the activation barriers are most pronounced in the condensate, and so is the concentration gradient. Equilibrium pertains at the surface, but the mass-transfer coefficient varies with time. Pressure in the vapor phase is constant. 

In the previous sections, stagnant films were assumed to exist on each side of the interface, and the normal mass transfer coefficients were assumed proportional to the first power of the molecular diffusivity. In many mass transfer operations, the rate of transfer varies with only a fractional power of the diffusivity because of flow in the boundary layer or because of the short lifetime of surface elements. The penetration theory is a model for short contact times that has often been applied to mass transfer from bubbles, drops, or moving liquid films. The equations for unsteady-state diffusion show that the concentration profile near a newly created interface becomes less steep with time, and the average coefficient varies with the square root of (D/t) [4] ... 

The Biot s criterion of mass transmission can reflect the resistance characteristic of gas diffusion field, that is the relative size between coal particle internal resistance and exterior convection mass transmission resistance, which is showed as attenuation coefficient on exterior the Fourier s criterion of mass transmission can represent the dynamic feature of gas diffusion field, that is the depth and scope of coal particle gas diffusion field varying with time, which is showed as gas initialization diffusion intensity of shearing fall coal particles on exterior. 

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. 

Horizontal diffusion. As the cloud expands horizontally, the particle size distribution may vary as a function of distance from the center if the diffusion coefficient is a function of particle size. This would cause the true correction factor to vary with horizontal position, and should be observable if observations are made for horizontal positions at the same altitude and time. Since the diffusion coefficient for clouds is known only within a factor of 10-100, no estimate of the variation with particle size is possible at this time. 

The theories vary in the assumptions and boundary conditions used to integrate Fick s law, but all predict the film mass transfer coefficient is proportional to some power of the molecular diffusion coefficient D", with n varying from 0.5 to 1. In the film theory, the concentration gradient is assumed to be at steady state and linear, (Figure 3-2) (Nernst, 1904 Lewis and Whitman, 1924). However, the time of exposure of a fluid to mass transfer may be so short that the steady state gradient of the film theory does not have time to develop. The penetration theory was proposed to account for a limited, but constant time that fluid elements are exposed to mass transfer at the surface (Higbie, 1935). The surface renewal theory brings in a modification to allow the time of exposure to vary (Danckwerts, 1951). 

On a RDE, in the absence of a surface layer, the EHD impedance is a function of a single dimensionless frequency, pSc1/3. This means that if the viscosity of the medium directly above the surface of the electrode and the diffusion coefficient of the species of interest are independent of position away from the electrode, then the EHD impedance measured at different rotation frequencies reduces to a common curve when plotted as a function of p. In other words, there is a characteristic dimensionless diffusional relaxation time for the system, pD, strictly (pSc1/3)D, which is independent of the disc rotation frequency. However, if v or D vary with position (for example, as a consequence of the formation of a viscous boundary layer or the presence of a surface film), then, except under particular circumstances described below, reduction of the measured parameters to a common curve is not possible. Under these conditions pD is dependent upon the disc rotation frequency. The variation of the EHD impedance with as a function of p is therefore the diagnostic for... 

In order to elucidate the PS diffusion behaviour in PMMA gel, the PFGStE aH NMR measurements have been made with varying values of the diffusion time A. For samples A1 and A2, the diffusion coefficients for the fast and slow diffusion components (as indicated by Dfast and Dsiow, respectively) and the corresponding fractions (/fast and/siow, respectively) are determined by using Equation (2), and Dfast and Dsiow are plotted in Figure 14A and B. The single diffusion coefficients of... 

Let us note the diffusion behaviour of small molecules, that is unreacted MMA, in the PMMA gel matrix. Using the vinyl peak of the unreacted MMA, PGSE 3H NMR measurements on the small molecule diffusion are performed for PMMA gel samples A1-A4 with varying A. The experimental data lie on a straight line in the A range from 60 to 500 ms, and the slope of the plots is independent of the diffusion time A. This clearly shows that the diffusion of the small molecule is a single mode and not restricted. The diffusion coefficient of MMA (D) as obtained from the slope of the straight line is also independent of the polymer concentration for samples A1-A4, D (10 9 m2 s ) 1.4,1.5,1.5 and 1.4, respectively. The... 

Equation 1.1 is commonly known as Fick s first law of diffusion, where Dj is the diffusion coefficient of species j. For Jj in mol m-2 s-1 and Cj in mol m-3 (hence, dc/dx in molm-4), Dj has units of m2s 1. Because Dj varies with concentration, temperature, and the medium for diffusion, it is properly called a coefficient in the general case. In certain applications, however, we can obtain sufficient accuracy by treating Dj as a constant. The partial derivative is used in Equation 1.1 to indicate the change in concentration in the x-direction of Cartesian coordinates at some moment in time (constant t) and for specified values of y and z. For most of the cases that we will consider, the flux density in... 

---