Molecular diffusion applications

Le Bihan, D., J. Delannoy, and R.L. Levin, Temperature mapping with MR imaging of molecular diffusion application to hyperthermia Radiology, 1989. 171(3) p. 853-7. 

Glaser and Litt (G4) have proposed, in an extension of the above study, a model for gas-liquid flow through a b d of porous particles. The bed is assumed to consist of two basic structures which influence the fluid flow patterns (1) Void channels external to the packing, with which are associated dead-ended pockets that can hold stagnant pools of liquid and (2) pore channels and pockets, i.e., continuous and dead-ended pockets in the interior of the particles. On this basis, a theoretical model of liquid-phase dispersion in mixed-phase flow is developed. The model uses three bed parameters for the description of axial dispersion (1) Dispersion due to the mixing of streams from various channels of different residence times (2) dispersion from axial diffusion in the void channels and (3) dispersion from diffusion into the pores. The model is not applicable to turbulent flow nor to such low flow rates that molecular diffusion is comparable to Taylor diffusion. The latter region is unlikely to be of practical interest. The model predicts that the reciprocal Peclet number should be directly proportional to nominal liquid velocity, a prediction that has been confirmed by a few determinations of residence-time distribution for a wax desulfurization pilot reactor of 1-in. diameter packed with 10-14 mesh particles. 

The molecular diffusivity D may be expressed in terms of the molecular velocity um and the mean free path of the molecules Xrn. In Chapter 12 it is shown that for conditions where the kinetic theory of gases is applicable, the molecular diffusivity is proportional to the product umXm. Thus, the higher the velocity of the molecules, the greater is the distance they travel before colliding with other molecules, and the higher is the diffusivity D. 

In many applications of mass transfer the solute reacts with the medium as in the case, for example, of the absorption of carbon dioxide in an alkaline solution. The mass transfer rate then decreases in the direction of diffusion as a result of the reaction. Considering the unidirectional molecular diffusion of a component A through a distance Sy over area A. then, neglecting the effects of bulk flow, a material balance for an irreversible reaction of order n gives ... 

Recent Uses of Solid-Surface Luminescence Analysis in Environmental Analysis. Vo-Dinh and coworkers have shown very effectively how solid-surface luminescence techniques can be used for environmentally important samples (17-22). RTF has been used for the screening of ambient air particulate samples (17,18). In addition, RTF has been employed in conjunction with a ranking index to characterize polynuclear aromatic pollutants in environmental samples (19). A unique application of RTF reported recently is a personal dosimeter badge based on molecular diffusion and direct detection by RTF of polynuclear aromatic pollutants (20). The dosimeter is a pen-size device that does not require sample extraction prior to analysis. 

In accordance with Pick s Law, diffusive flow always occurs in the direction of decreasing concentration and at a rate, which is proportional to the magnitude of the concentration gradient. Under true conditions of molecular diffusion, the constant of proportionality is equal to the molecular diffusivity of the component i in the system, D, (m /s). For other cases, such as diffusion in porous matrices and for turbulent diffusion applications, an effective diffusivity value is used, which must be determined experimentally. 

Fabrication processing of these materials is highly complex, particularly for materials created to have interfaces in morphology or a microstructure [4—5], for example in co-fired multi-layer ceramics. In addition, there is both a scientific and a practical interest in studying the influence of a particular pore microstructure on the motional behavior of fluids imbibed into these materials [6-9]. This is due to the fact that the actual use of functionalized ceramics in industrial and biomedical applications often involves the movement of one or more fluids through the material. Research in this area is therefore bi-directional one must characterize both how the spatial microstructure (e.g., pore size, surface chemistry, surface area, connectivity) of the material evolves during processing, and how this microstructure affects the motional properties (e.g., molecular diffusion, adsorption coefficients, thermodynamic constants) of fluids contained within it. 

Here, A is the random path, B the longitudinal molecular diffusion and C the RTMT contributions with the velocity of the mobile phase u shown separately. H is referred to as the Height Equivalent to a Theoretical Plate and is terminology borrowed from distillation. While the distillation HETP is not truly applicable, the terminology has persisted. It can be shown that the H in this expression is the equivalent to variance/unit length. This is the expression introduced by Van Deemter and co-workers in 1956 in a discussion of band broadening. 

As discussed in Chapter 5, the complexity of the chemical source term restricts the applicability of closures based on second- and higher-order moments of the scalars. Nevertheless, it is instructive to derive the scalar covariance equation for two scalars molecular-diffusion coefficients ra and I, respectively. Starting from (1.28), p. 16, the transport equation for ((,) can be found following the same steps that were used for the Reynolds stresses. This process yields34... 

In this case, if the boundary and initial conditions allow it, either ej or c can be used to define the mixture fraction. The number of conserved scalar transport equations that must be solved then reduces to one. In general, depending on the initial conditions, it may be possible to reduce the number of conserved scalar transport equations that must be solved to min(Mi, M2) where M = K - Nr and M2 = number of feed streams - 1. In many practical applications of turbulent reacting flows, M =E and M2 = 1, and one can assume that the molecular-diffusion coefficients are equal thus, only one conserved scalar transport equation (i.e., the mixture fraction) is required to describe the flow. 

We will now describe the application of the two principal methods for considering mass transport, namely mass-transfer models and diffusion models, to PET polycondensation. Mass-transfer models group the mass-transfer resistances into one mass-transfer coefficient ktj, with a linear concentration term being added to the material balance of the volatile species. Diffusion models employ Fick s concept for molecular diffusion, i.e. J = — D,v ()c,/rdx, with J being the molar flux and D, j being the mutual diffusion coefficient. In this case, the second derivative of the concentration to x, DiFETd2Ci/dx2, is added to the material balance of the volatile species. 

In addition to phase change and pyrolysis, mixing between fuel and oxidizer by turbulent motion and molecular diffusion is required to sustain continuous combustion. Turbulence and chemistry interaction is a key issue in virtually all practical combustion processes. The modeling and computational issues involved in these aspects have been covered well in the literature [15, 20-22]. An important factor in the selection of sub-models is computational tractability, which means that the differential or other equations needed to describe a submodel should not be so computationally intensive as to preclude their practical application in three-dimensional Navier-Stokes calculations. In virtually all practical flow field calculations, engineering approximations are required to make the computation tractable. 

As already said, Taylor s effective model contains a contribution in the effective diffusion coefficient, which is proportional to the square of the transversal Peclet number. Frequently this term is more important than the original molecular diffusion. After his work, it is called Taylor s dispersion coefficient and it is generally accepted and used in chemical engineering numerical simulations. For the practical applications we refer to the classical paper (Rubin, 1983) by Rubin. The mathematical study of the models from Rubin (1983) was undertaken in Friedman and Knabner (1992). 

When pulsed magnetic field gradients are applied to study diffusive processes, the MR technique is often referred to as pulsed field gradient or pulsed gradient spin echo (PGSE) MR. Application of PGSE MR techniques to quantify molecular diffusion was pioneered by Stejskal and Tanner 17,18), and the techniques typically probe molecular displacements of 10 -10 m over time scales of the order 10 M s. 

The shear rate (s ) at the membrane surface is given by = 9ivld (cf. Example 2.2), where v is the average linear velocity (cm s ), d is the fiber inside diameter (cm), and Dp is the molecular diffusion coefficient (cm s ) all other symbols are as in Equation 8.4. Hence, Equation 8.6 is most likely applicable to UF in hollow-fiber membranes in general. 

The measurement of a molecular diffusion coefficient D by electrochemical techniques is generally done with a rotating disk electrode in the limiting diffusion current condition and application of the Levich s equation [8]. 

Kehr M, Fatkullin N, Kimmich R (2007) Molecular diffusion on a time scale between nano-and milliseconds probed by field-cycling NMR relaxometry of intermolecular dipolar interactions Application to polymer melts. J Chem Phys 126 094903 Kirkwood JG, Riseman J (1948) The intrinsic viscosity and diffusion constant of flexible macromolecules in solution. J Chem Phys 16 565-573 Klein J (1986) Dynamics of entangled linear, branched, and cyclic polymers. Macromolecules 19(1) 105—118... 

---