Mass transport molecular diffusion coefficient

The dimensionless form of the equation contains one dimensionless parameter as a multiplier of the first term of the right-hand side and maybe some additional dimensionless parameters, which may appear within the dimensionless source term, S. Depending on the general variable, 0, the effective diffusion coefficient, F, appearing in this dimensionless number will be different, leading to different dimensionless numbers. For the species mass fraction, momentum and enthalpy transport equations, the effective diffusion coefficient will be molecular diffusion coefficient, the kinematic viscosity of the fluid and the thermal diffusivity of the fluid respectively. The corresponding dimensionless numbers are, therefore, defined as follows. 

Equation [3-17] does not hold at very low seepage velocities because mechanical dispersion no longer dominates Fickian mass transport. When the mechanical dispersion coefficient becomes less than the effective molecular diffusion coefficient, the longer travel times associated with lower velocities do not result in further decreases in Fickian mass transport. 

A schematic representation of the boundary layers for momentum, heat and mass near the air—water interface. The velocity of the water and the size of eddies in the water decrease as the air—water interface is approached. The larger eddies have greater velocity, which is indicated here by the length of the arrow in the eddy. Because random molecular motions of momentum, heat and mass are characterized by molecular diffusion coefficients of different magnitude (0.01 cm s for momentum, 0.001 cm s for heat and lO cm s for mass), there are three different distances from the wall where molecular motions become as important as eddy motions for transport. The scales are called the viscous (momentum), thermal (heat) and diffusive (molecular) boundary layers near the interface. 

If a fluid is placed between two concentric cylinders, and the iimer cylinder rotated, a complex fluid dynamical motion known as Taylor-Couette flow is established. Mass transport is then by exchange between eddy vortices which can, under some conditions, be imagined as a substantially enhanced diffiisivity (typically with effective diffusion coefficients several orders of magnitude above molecular diffusion coefficients) that can be altered by varying the rotation rate, and with all species having the same diffusivity. Studies of the BZ and CIMA/CDIMA systems in such a Couette reactor [45] have revealed bifurcation through a complex sequence of front patterns, see figure A3.14.16. 

The binary molecular diffusion coefficient, ab> has units of length /time and characterizes the microscopic motion of species A in solvent B, for example. Hab is also the molecular transport property that appears in the linear law that relates diffusional fluxes and concentration gradients. In this respect, the same quantity, Bab. represents a molecular transport property for mass transfer and a diffusion coefficient. This is not the case for the other two transport processes. 

The mass transfer-chemical reaction process occurs isothermally. This critical assumption allows one to neglect energy transport processes within the reactor. Hence, the physical properties of the reactive mixture—overall mass density, viscosity, binary molecular diffusion coefficient, and surface-averaged kinetic rate constant—are treated as constants throughout the reactor. 

A fluid passing over a solid surface develops a boundary layer in which the velocity parallel to the surface varies rapidly over a very short distance normal to the direction of flow. The velocity is zero at the solid surface but approaches the bulk-stream velocity at a distance less than a millimeter from the surface. Mixing occurs in the main fluid stream where reactants and products are transported at rates that depend on the nature of flow. The fluid velocity near the surface is low with little mixing therefore, mass transport perpendicular to the surface is by molecular diffusion. Although mass transport in the main stream is essentially independent of the molecular diffusion coefficient, it is proportional to DA-mix very... 

For weU-defined reaction zones and irreversible, first-order reactions, the relative reaction and transport rates are expressed as the Hatta number, Ha (16). Ha equals (k- / l ) where k- = reaction rate constant, = molecular diffusivity of reactant, and k- = mass-transfer coefficient. Reaction... 

Diffusion is the molecular transport of mass without flow. The diffu-sivity (D) or diffusion coefficient is the proportionality constant between the diffusion and the concentration gradient causing diffusion. It is usually defined by Fick s first law for one-dimensional, binary component diffusion for molecular transport without turbulence shown by Eq. (2-149)... 

The modeling of mass transport from the bulk fluid to the interface in capillary flow typically applies an empirical mass transfer coefficient approach. The mass transfer coefficient is defined in terms of the flux and driving force J = kc(cbuik-c). For non-reactive steady state laminar flow in a square conduit with constant molecular diffusion D, the mass balance in the fluid takes the form... 

Diffusion of small molecular penetrants in polymers often assumes Fickian characteristics at temperatures above Tg of the system. As such, classical diffusion theory is sufficient for describing the mass transport, and a mutual diffusion coefficient can be determined unambiguously by sorption and permeation methods. For a penetrant molecule of a size comparable to that of the monomeric unit of a polymer, diffusion requires cooperative movement of several monomeric units. The mobility of the polymer chains thus controls the rate of diffusion, and factors affecting the chain mobility will also influence the diffusion coefficient. The key factors here are temperature and concentration. Increasing temperature enhances the Brownian motion of the polymer segments the effect is to weaken the interaction between chains and thus increase the interchain distance. A similar effect can be expected upon the addition of a small molecular penetrant. 

Ordinary or bulk diffusion is primarily responsible for molecular transport when the mean free path of a molecule is small compared with the diameter of the pore. At 1 atm the mean free path of typical gaseous species is of the order of 10 5 cm or 103 A. In pores larger than 1CT4 cm the mean free path is much smaller than the pore dimension, and collisions with other gas phase molecules will occur much more often than collisions with the pore walls. Under these circumstances the effective diffusivity will be independent of the pore diameter and, within a given catalyst pore, ordinary bulk diffusion coefficients may be used in Fick s first law to evaluate the rate of mass transfer and the concentration profile in the pore. In industrial practice there are three general classes of reaction conditions for which the bulk value of the diffusion coefficient is appropriate. For all catalysts these include liquid phase reactions... 

Diffusion is characterized by a mass transfer coefficient U8 of 104 m/h, which can be regarded as a molecular diffusivity of 2 x 10 6 m2/h divided by a path length of 0.02 m. In practice, bioturbation may contribute substantially to this exchange process, and in shallow water current-induced turbulence may also increase the rate of transport. Diffusion in association with organic colloids is not included. The D value is thus given as Us AwZ2. 

Dispersion in packed tubes with wall effects was part of the CFD study by Magnico (2003), for N — 5.96 and N — 7.8, so the author was able to focus on mass transfer mechanisms near the tube wall. After establishing a steady-state flow, a Lagrangian approach was used in which particles were followed along the trajectories, with molecular diffusion suppressed, to single out the connection between flow and radial mass transport. The results showed the ratio of longitudinal to transverse dispersion coefficients to be smaller than in the literature, which may have been connected to the wall effects. The flow structure near the wall was probed by the tracer technique, and it was observed that there was a boundary layer near the wall of width about Jp/4 (at Ret — 7) in which there was no radial velocity component, so that mass transfer across the layer... 

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. 

The negative sign designates that the direction of flow occurs spontaneously from the region of higher concentration. The diffusion coefficient D represents the amount of solute that diffuses across a 1 cm area per sec under the influence of a concentration gradient of 1 g/cm. D is characteristic for each type of solute molecule, and its value takes into account both molecular mass and shape. See also Transport Processes Pick s Laws... 

Turbulent diffusion occurs because turbulent eddies are transporting mass, momentum, and energy over the eddy scale at the rotational velocity. This transport rate is generally orders of magnitude greater than the transport rate due to molecular motion. Thus, when a flow is turbulent, diffusion is normally ignored because e Z). The exception is very near the flow boundaries, where the eddy size (and turbulent diffusion coefficient) decreases to zero. 

The Molecular Origins of Mass Diffusivity. In a manner directly analogous to the derivations of Eq. (4.6) for viscosity and Eq. (4.34) for thermal conductivity, the diffusion coefficient, or mass diffusivity, D, in units of m /s, can be derived from the kinetic theory of gases for rigid-sphere molecules. By means of summary, we present all three expressions for transport coefficients here to further illustrate their similarities. 

Deacon s intention was to separate the viscosity effect from the wind effect, so that the new model would be able to describe the change of via due to a change of water or air temperature (i.e., of viscosity) at constant wind speed. Deacon concluded that mass transfer at the interface must be controlled by the simultaneous influence of two related processes, that is, by the transport of chemicals (described by molecular diffusivity Dia), and by the transport of turbulence (described by the coefficient of kinematic viscosity va). Note that v has the same dimension as Dia. Thus, the ratio between the two quantities is nondimensional. It is called the Schmidt Number, Scia ... 

In this text we are concerned exclusively with laminar flows that is, we do not discuss turbulent flow. However, we are concerned with the complexities of multicomponent molecular transport of mass, momentum, and energy by diffusive processes, especially in gas mixtures. Accordingly we introduce the kinetic-theory formalism required to determine mixture viscosity and thermal conductivity, as well as multicomponent ordinary and thermal diffusion coefficients. Perhaps it should be noted in passing that certain laminar, strained, flames are developed and studied specifically because of the insight they offer for understanding turbulent flame environments. 

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