Diffusive Transport Example

A simple example of molecular transport for a three-species problem is illustrated here. Consider evaporation of a liquid into a high-aspect-ratio tube open to air. The diffusive transport of species entering the vapor through evaporation can be solved as a onedimensional two-point boundary-value problem. 

A schematic of the experiment is shown in Fig. 12.14. The level of the liquid is maintained (in some manner) at a constant height, arbitrarily denoted z = 0. In addition the liquid is well mixed and maintained at constant solute concentration. For this example, the liquid considered is a 30% solution of HC1 in water, evaporating into air (both liquid and vapor at 20°C). Air will be considered as a third species, rather than treating air s chemical components (oxygen, nitrogen, etc.) separately. For ease of notation, the species will be referred to by number as 1 = HC1,2 = H2O, and 3 = Air. 

A stream of dry air blows across the open top of the tube. Thereby the concentrations (or equivalently the mole fractions) of HC1 and H2O at the z = Z are assumed to drop to zero. For this example, assume a tube height of Z = 0.1 m, open to atmospheric pressure (i.e, p = 101325 Pa). The mole fractions of HC1 and H2O at the liquid-vapor interface are assumed to be at their equilibrium values, 0j01395, and 0.00712, respectively [312]. 

At steady state, the molar fluxes of HC1 and H2O are each constants at all heights along the tube the flux of air in the column at steady state must be zero. We seek a solution for the molar flux of HC1 and H2O consistent with the mole fraction boundary conditions at the surface and at the top of the tube specified above. Thus this is a two-point boundary- 

These equations are written in terms of the (laboratory referenced) velocities V. However, from analysis of the problem it is clear that the molar flux of each species is constant, and this knowledge can be used in formulation of the governing equations. 

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In the case of systems containing ionic liquids, components and chemical species have to be differentiated. The methanol/[BMIM][PF6] system, for example, consists of two components (methanol and [BMIM][PFg]) but - on the assumption that [BMIM][PFg] is completely dissociated - three chemical species (methanol, [BMIM] and [PFg] ). If [BMIM][PFg] is not completely dissociated, one has a fourth species, the undissociated [BMIM][PFg]. From this it follows that the diffusive transport can be described with three and four flux equations, respectively. The fluxes of [BMIM] ... 

The excess electrolyte is often termed the indifferent electrolyte. From the practical point of view, solutions containing an indifferent electrolyte are very often used in miscellaneous investigations. For example, when determining equilibrium constants (e.g. apparent dissociation constants, Eq. 1.1.26) it is necessary only to indicate the indifferent electrolyte and its concentration, as they do not change when the concentrations of the reactants are changed. Moreover, the indifferent electrolyte is important in the study of diffusion transport (Section 2.5), for elimination of liquid... 

Another example is linear diffusion, with a prescribed concentration gradient at the reference plane, i.e. a prescribed material flux through the reference plane. This type of diffusion transport is important mainly for electrode processes (see Section 5.4). The point of interest in this case is the concentration at the reference plane. In the simplest case, the material flux is constant, so that the boundary condition for x = 0 (Eq. 2.5.5) can be replaced by... 

Thus the addition of an inert gas which does not intervene chemically in the transport reaction but adds to the density of the gas, reduces the segregation due to thermal diffusion. An example of this is the reduction of thermal separation in a mixture of H2 and H20 by the addition of Hg vapour (Dastur and Chipman, 1948). 

In natural waters, unattached microorganisms move with the bulk fluid [55], so that no flux enhancement will occur due to fluid motion for the uptake of typical (small) solutes by small, freely suspended microorganisms [25,27,35,41,56,57], On the other hand, swimming and sedimentation have been postulated to alleviate diffusive transport limitation for larger organisms. Indeed, in the planar case (large r0), the diffusion boundary layer, 8, has been shown to depend on advection and will vary with D according to a power function of Da (the value of a is between 0.3 and 0.7 [43,46,58]). For example, in Chapter 3, it was demonstrated that in the presence of a laminar flow parallel to a planar surface, the thickness of the diffusion boundary layer could be estimated by ... 

Figure 1.4 gives an example of the adsorption of a compound to suspended sediment, modeled as two resistances in series. At first, the compound is dissolved in water. For successful adsorption, the compound must be transported to the sorption sites on the surface of the sediment. The inverse of this transport rate can also be considered as a resistance to transport, Ri. Then, the compound, upon reaching the surface of the suspended sediment, must find a sorption site. This second rate parameter is more related to surface chemistry than to diffusive transport and is considered a second resistance, R2, that acts in series to the first resistance. The second resistance cannot... 

Equation (56) states that the effect of a thermal gradient on the material transport bears a reciprocal relationship to the effect of a composition gradient upon the thermal transport. Examples of Land L are the coefficient of thermal diffusion (S19) and the coefficient of the Dufour effect (D6). The Onsager reciprocity relationships (Dl, 01, 02) are based upon certain linear approximations that have a firm physical foundation only when close to equilibrium. For this reason it is possible that under circumstances in which unusually high potential gradients are encountered the coupling between mutually related effects may be somewhat more complicated than that indicated by Eq. (56). Hirschfelder (BIO, HI) discussed many aspects of these cross linkings of transport phenomena. 

There are, however, various types of active transport systems, involving protein carriers and known as uniports, symports, and antiports as indicated in Figure 3.7. Thus, symports and antiports involve the transport of two different molecules in either the same or a different direction. Uniports are carrier proteins, which actively or passively (see section "Facilitated Diffusion") transport one molecule through the membrane. Active transport requires a source of energy, usually ATP, which is hydrolyzed by the carrier protein, or the cotransport of ions such as Na+ or H+ down their electrochemical gradients. The transport proteins usually seem to traverse the lipid bilayer and appear to function like membrane-bound enzymes. Thus, the protein carrier has a specific binding site for the solute or solutes to be transferred. For example, with the Na+/K+ ATPase antiport, the solute (Na+) binds to the carrier on one side of... 

Materials processing, via approaches like chemical vapor deposition (CVD), are important applications of chemically reacting flow. Such processes are used widely, for example, in the production of silicon-based semiconductors, compound semiconductors, optoelectronics, photovoltaics, or other thin-film electronic materials. Quite often materials processing is done in reactors with reactive gases at less than atmospheric pressure. In this case, owing to the fact that reducing pressure increases diffusive transport compared to inertial transport, the flows tend to remain laminar. 

Frequently, a major limitation of DS-based collection systems is that they operate at substantially subquantitative collection efficiencies at the typical sampling rates used. This situation increases the probability of error because of large thermal variations that affect diffusive transport. For these reasons, should wet denuders (vide infra) prove to be viable continuous collection devices, they may well replace DS-based systems. Their ability to more quantitatively remove gases may also spur the development of combined gas-particle analyzer systems in which, for example, the acid gases are removed by the denuder and analyzed the particles are then collected by the impactor equivalent of a wet denuder, and the acidity associated with... 

Examples of dimensionless groups that specify ratios of transport mechanisms are listed next in Table II and depend on the size and shape of the domain. The Peclet numbers for heat (Pet) and solute (Pes) and momentum (Re) transport are ratios of scales for convective to diffusive transport and depend on the magnitudes of the velocity field and the length scale for the diffusion gradient. Boundary layers form at large Peclet numbers (Pet or Pes) or Reynolds numbers (Re). The fonnation of a boundary layer at a large Re is particularly important in crystal growth from the melt, because the low... 

Ester prodrugs are employed to enhance membrane permeation and transepithelial transport of hydrophilic drugs by increasing the lipophilicity of the parent compound, resulting in enhanced transmembrane transport by passive diffusion. For example, pivampicillin, a pival-oyloxymethyl ester of ampicillin, is more lipophilic than its parent ampicillin and has demonstrated increased membrane permeation and transepithelial transport in in vivo studies.103... 

Diffusion is the transport of a chemical by random motion due to a state of disequilibrium. For example, diffusion causes the movement of a chemical within a phase (e.g., water) from a location of relatively high concentration to a place of lower concentration until the chemical is homogeneously distributed throughout the phase. Likewise diffusive transport will drive a chemical between media (e.g., water and air) until their equilibrium concentrations are reached and thus the chemical potentials or fugacities are equal in each phase. 

The construction of a mass balance model follows the general outline of this chapter. First, one defines the spatial and temporal scales to be considered and establishes the environmental compartments or control volumes. Second, the source emissions are identified and quantified. Third, the mathematical expressions for advective and diffusive transport processes are written. And last, chemical transformation processes are quantified. This model-building process is illustrated in Figure 27.4. In this example we simply equate the change in chemical inventory (total mass in the system) with the difference between chemical inputs and outputs to the system. The inputs could include numerous point and nonpoint sources or could be a single estimate of total chemical load to the system. The outputs include all of the loss mechanisms transport... 

Figure 11.1 Schematic examples of passive diffusion, facilitated transport and coupled transport. The facilitated transport example shows permeation of oxygen across a membrane using hemoglobin as the carrier agent. The coupled transport example shows permeation of copper and hydrogen ions across a membrane using a reactive oxime as the carrier agent...

The simplest of these functions is that of a permeability barrier that limits free diffusion of solutes between the cytoplasm and external environment. Although such barriers are essential for cellular life to exist, there must also be a mechanism by which selective permeation allows specific solutes to cross the membrane. In contemporary cells, such processes are carried out by transmembrane proteins that act as channels and transporters. Examples include the proteins that facilitate the transport of glucose and amino acids into the cell, channels that allow potassium and sodium ions to permeate the membrane, and active transport of ions by enzymes that use ATP as an energy source. 

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