Diffusion coefficient temperature dependence

Let us assume a spherical mineral with radius R which initially contains a gas with concentration C0(r), r being the radial distance from the center. Upon incremental heating, this gas is lost to the extraction line and at the ith heating step when time is tf, the fraction of initial gas remaining is/(tf). Loss takes place by radial diffusion with temperature-dependent, hence time-dependent, coefficient 3>(t). We assume that the total amount of gas held by the mineral at t=0 is equal to one, i.e., that... 

Up to now, only hydrodynamic repulsion effects (Chap. 8, Sect. 2.5) have caused the diffusion coefficient to be position-dependent. Of course, the diffusion coefficient is dependent on viscosity and temperature [Stokes—Einstein relationship, eqn. (38)] but viscosity and temperature are constant during the duration of most experiments. There have been several studies which have shown that the drift mobility of solvated electrons in alkanes is not constant. On the contrary, as the electric field increases, the solvated electron drift velocity either increases super-linearly (for cases where the mobility is small, < 10 4 m2 V-1 s-1) or sub-linearly (for cases where the mobility is larger than 10 3 m2 V 1 s 1) as shown in Fig. 28. Consequently, the mobility of the solvated electron either increases or decreases, respectively, as the electric field is increased [341— 348]. 

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... 

The temperature dependence of the diffusion coefficient will depend on the diffusion mechanism. If diffusion occurs interstitially, the temperature dependence of D will include only the migration energy term, Ai/, since the probability of the site adjacent to an interstitial atom being vacant is 1.0. 

Although how the thermal diffusion coefficient Dx depends on particle size is still under debate, thermophoresis has been applied for the separation of particles with different sizes in recent decades. In addition, as thermophoresis is dependent on molecular size, surface charge, and hydration shell (the result of surrounding water molecules), it is used to analyze protein functionality and the interactions of proteins or other molecules in biological liquids such as blood serum or cell lysate. For example, through measuring the thermal diffusion coefficients Dt of RNAs and DNAs at various temperatures, the conformation and stability of RNA and DNA can be quantified. 

The temperature dependence of a diffusion coefficient is determined by the activation energy of diffusion. The temperature dependence of the solubility coefficient is given by the enthalpy of solution AH. Hence, the temperature dependence of the permeability coefficient is given by... 

In this case the molecules collide with the walls of the pores more frequently than with each other (Knudsen diffusion) and the diffusion coefficient Dk depends not only on the absolute temperature, T, but also on the radius of the pores (rp), i.e., Dk = K rp V(T/Mi), Mi being the molecular weight of the diffusing species (k = 9700 if rp is expressed in cm). 

If the temperature cannot be assumed constant, then the equations have to be solved numerically. The same is true if the diffusion coefficient is dependent on the concentration. 

If the polymer is above its glass transition tempetature, Tg, it responds rapidly to changes in its physical condition and we have Fickian or Case I diffusion. This is the simplest case, and for T>T, Henry s law is valid for sorption and the diff ion coefficient is a constant (ideal Fickian diffusion). Its temperature dependence is well approximated by a simple Arrhenius expression with a constant activation energy. 

Keywords diffusion through polymeric films, diffusion coefficient temperature and concentration dependence, modelling of diffusion, sorption, crystallinity. 

The ternary diffusion coefficient strongly depends on the solution concentration. In order to calculate accurate mass transfer coefficients, experimental data of diffusion coefficients at the interest concentrations and temperatures are necessary. However, data are not available at concentrations and temperature used at the present study, it was assumed that the ternary diffusion coefficients were equal to the binary diffusion coefficients. The binary diffusion coefficients of the KDP-water pairs and the urea-water pairs were taken from literature (Mullin and Amatavivadhana, 1967 Cussler, 1997). The values were transformed into the Maxwell-Stefan diffiisivities using the thermodynamic correction factor. 

X 10 m /s. Diffusion coefficients may be corrected for other conditions by assuming them proportional to Schmidt numbers depend only weaMy on temperature (113). 

Other Properties. The glass-transition temperature for PPO is 190 K and varies htde with molecular weight (182). The temperature dependence of the diffusion coefficient of PPO in the undiluted state has been measured (182). 

The temperature dependence of the permeability arises from the temperature dependencies of the diffusion coefficient and the solubility coefficient. Equations 13 and 14 express these dependencies where and are constants, is the activation energy for diffusion, and is the heat of solution... 

Frequently, the transport coefficients, such as diffusion coefficient orthermal conductivity, depend on the dependent variable, concentration, or temperature, respectively. Then the differential equation might look Bke... 

Hayduk-Laudie They presented a simple correlation for the infinite dilution diffusion coefficients of nonelectrolytes in water. It has about the same accuracy as the Wilke-Chang equation (about 5.9 percent). There is no explicit temperature dependence, but the 1.14 exponent on I compensates for the absence of T in the numerator. That exponent was misprinted (as 1.4) in the original article and has been reproduced elsewhere erroneously. 

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