Diffusivity, mass coefficients

N. C. Bartelt, T. L. Einstein, E. D. Williams. Measuring surface mass diffusion coefficients by observing step fluctuation. Surf Sci 572 411, 1994. 

Applied Pressure Equilibrium Mass Diffusion Coefficient D... 

The dynamical properties of polymer molecules in solution have been investigated using MPC dynamics [75-77]. Polymer transport properties are strongly influenced by hydrodynamic interactions. These effects manifest themselves in both the center-of-mass diffusion coefficients and the dynamic structure factors of polymer molecules in solution. For example, if hydrodynamic interactions are neglected, the diffusion coefficient scales with the number of monomers as D Dq /Nb, where Do is the diffusion coefficient of a polymer bead and N), is the number of beads in the polymer. If hydrodynamic interactions are included, the diffusion coefficient adopts a Stokes-Einstein formD kltT/cnr NlJ2, where c is a factor that depends on the polymer chain model. This scaling has been confirmed in MPC simulations of the polymer dynamics [75]. 

On the basis of scaling arguments, general functional dependencies can also be derived. For example, dimensional analysis shows that the center of mass diffusion coefficient DG for Zimm relaxation has the form... 

The prediction of a time-dependent centre of mass diffusion coefficient has recently been corroborated by a combined atomistic simulation and an NSE approach on PB ([55]). The dynamic structure factor from simulation and experiment obtained at 353 K are displayed in Fig. 3.11. 

An analytical ultracentrifugation method for determining the molecular mass, diffusion coefficient, and/or state of oligomerization of a macromolecule by conducting sedimentation conditions to establish an equilibrium distribution of the macromolecule from the meniscus to the bottom of the observation cell. 

The mass diffusivity coefficient of isobutane blowing agent from LDPE foam was found using a onedimensional diffusion model of two concentric cylinders with Dirichlet boundary conditions. An average mass diffusivity coefficient was used to calculate the mass of isobutane remaining in the foam for different boundary conditions. The influence of temperature and additives on diffusion was also examined. The use of the mass diffusivity coefficient in assessing the flammability of PE foam in the post-extrusion period is discussed. 2 refs. USA... 

If the mass diffusion coefficient is sufficiently large compared with the thermal diffusivity, so P > / c, the range between n2- and n+ will be non-zero. There is another consideration n can only have discrete integer values, the lowest of which for a non-uniform state is n = 1. Thus for observable patterns we must make sure that at least n exceeds unity. Equation (10.49) shows that this last requirement puts a lower bound on the size parameter y we need y > yc, where... 

Dj is the mass diffusion coefficient, and cgas is the total molar concentration of the gas mixture. Although Equations (3.9a) and (3.9b) can be used for a free-path gas (e.g. gas channel), when a gas is moving within a porous media (i.e. electrode), Equation (3.9) may not be the most appropriate. Different constitutive laws can be employed for describing the diffusive flux within a porous medium. The choice of the most appropriate law depends on the operating conditions and the porous media properties, as further explained in Section 3.3.2. 

S)AB Binary mass diffusion coefficient in Fick s law (Example 2.4)... 

The dimensionless retention parameter X of all FFF techniques, if operated on an absolute basis, is a function of the molecular characteristics of the compounds separated. These include the size of macromolecules and particles, molar mass, diffusion coefficient, thermal diffusion coefficient, electrophoretic mobility, electrical charge, and density (see Table 1, Sect. 1.4.1.) reflecting the wide variablity of the applicable forces [77]. For detailed theoretical descriptions see Sects. 1.4.1. and 2. For the majority of operation modes, X is influenced by the size of the retained macromolecules or particles, and FFF can be used to determine absolute particle sizes and their distributions. For an overview, the accessible quantities for the three main FFF techniques are given (for the analytical expressions see Table l,Sect. 1.4.1) ... 

In the third simulation example, we carried out an analysis of some of the aspects that characterize the case of the mass transfer of species through a membrane which is composed of two layers (the separative and the support layers) with the same thickness but with different diffusion coefficients of each entity or species. To answer this new problem the early model has been modified as follows (i) the term corresponding to the source has been eliminated (u) different conditions for bottom and top surfaces have been used for example, at the bottom surface, the dimensionless concentration of species is considered to present a unitary value while it is zero at the top surface (iii) a new initial condition is used in accordance with this case of mass transport through a two-layer membrane (iv) the values of the four thermal diffusion coefficients from the original model are replaced by the mass diffusion coefficients of each entity for both membrane layers (v) the model is extended in order to respond correctly to the high value of the geometric parameter 1/L. 

The Lu number is the ratio between the mass diffusion coefficient and the heat diffusion coefficient. It can be interpreted as the ratio between the propagation velocity of the iso-concentration surface and the isothermal surface. In other words, it characterizes the inertia of the temperature field inertia, with respect to the moisture content field (the heat and moisture transfers inertia number). The LUp diffusive filtration number is the ratio between the diffusive filtration field potential (internal pressure field potential) and the temperature field propagation. 

ThFFF elution volume (or time) is a function of D y/D, where Dt is the thermal diffusion coefficient and D is the mass diffusion coefficient (9,10). The mass diffusion coefficient D of a polymer molecule in a fluid with viscosity rjo is given by (ii)... 

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... 

Here, D is the (mass) diffusion coefficient, U is the field-induced velocity of the sample, and w is the channel thickness. In thermal FFF, U is governed by the thermal diffusion coefficient i r) and the temperature gradient (dT/dx), which is applied in the same dimension (x) as the channel thickness (x varies in value from 0 at the cold wall to w at the hot wall). Using the dependence of U on Dj and dT/dx, the retention parameter in thermal FFF can be expressed as... 

In thermal FFF, the applied held is a temperature drop (AT) across the channel, and the physicochemical parameter that governs retention is the Soret coefficient, which is the ratio of the thermodiffusion coefficient Dj) to the ordinary (mass) diffusion coefficient D). Because AT is set by the user, retention in a thermal FFF channel can be used to calculate the Soret coeffident of a polymer-solvent system. 

Here, AT is the temperature drop across the channel, which is set by the user, and D is the ordinary (mass) diffusion coefficient. The parameter V° is the geometric volume of the channel, which is constant for a given instrument. Note that V, is the same parameter used to define retention in SEC and that the ratio on the left side of Eq. (1) is the number of channel volumes required to flush a sample component through the thermal FFF channel. Although Eq. (1) is an approximation, it becomes accurate to within 3% when V/V° > 10. More important for this discussion, Eq. (1) characterizes the influence of analyte parameters D and on retention in thermal FFF. 

The three transport properties of the greatest concern are the viscosity, thermal conductivity and mass-diffusion coefficients. In each case, although measurements had been conducted over a period of at least 150 years, it was not until around 1970 that techniques of an acceptable accuracy were developed for the relatively routine measurement of any of these properties. There is ample evidence in the literature of very large discrepancies among measurements made prior to that date. One reason for these discrepancies lies in the conflicting requirements that, to make a transport-property measurement, one must perturb an equilibrium state but, at the same... 

Dsr generalized non-symmetric multi-component Fickian mass diffusion coefficients nr /s)... 

Doulia, D., Tzia, K., and Gekas, V. A knowledge base for the apparent mass diffusion coefficient (Deff) of foods, Int. J. Food Prop., 3,1, 2000. 

Dh equals about 1CT7 m2 s-1 for most food components this is correct within a factor of two. For crystalline material the value tends to be higher than for a liquid, and ice even has Du x 10 6 m2 s 1 (most metals are in the range 10 5 10 4). Since the mass diffusion coefficient nearly always is < 10 9 m2 s, the diffusion of heat is at least 100 times as fast as the diffusion of mass. Nevertheless, it is still slow at distances longer than a few mm. 

Heat. Heat can also be transported by diffusion, also in solids. The diffusion coefficient then is called thermal diffusivity it has a fairly constant value that is much larger than that of mass diffusion coefficients. This means that temperature evens out much faster than concentration. To calculate the transport of the amount of heat, the diffusion coefficient in Fick s laws must be replaced by the thermal conductivity. Under various conditions, heat can also be transported by mixing, by radiation, and by distillation. 

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