Diffusive Flux Processes

Diffnsive flnx refers to material transport in response to a chemical potential gradient (concentration gradient) in a medium (gas or water or solids) that is stationary. If the medium is turbulent, then the transport of material is referred to as turbulent or eddy diffnsion (Herman, 1979). The flux (J ) of solutes diffusing through stationary gas or liquid medium is proportional to the concentration gradient (dC/dx)  

FIGURE 14.5 Schematic showing irregular path for solutes diffusing through soils. 

Diffusion Coefficients in Water (D = 10 cm s ) for Ions of Interest to Wetlands and Aquatic Systems 

Porosity dictates the extent of tortuous diffusive path. Soils of low porosity exhibit longer diffusive paths than those with high porosity (Sweerts et al., 1991). Boudreau (1997) suggested the following empirical relationship between tortuosity and porosity to estimate tortuosity factor. 

The diffusion coefficient of sediment or soil, in terms of tortuosity is expressed as  

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The tme driving force for any diffusive transport process is the gradient of chemical potential rather than the gradient of concentration. This distinction is not important in dilute systems where thermodynamically ideal behavior is approached. However, it becomes important at higher concentration levels and in micropore and surface diffusion. To a first approximation the expression for the diffusive flux may be written... 

It is a typical feature of the diffusion processes at electrodes of small size, which are reached by converging diffusion fluxes, that a steady state can be attained even without convection (e.g., in gelled solutions). Such electrodes, which have dimensions comparable to typical values of 8, are called microelectrodes. 

Thus, the problem is similar to the feedback mode case except for the internal boundary condition described by Eqs. (31) and (32), which relate to the first-order process at the target interface. The internal boundary condition describes the net diffusive flux of Red across the interface from phase 2 to phase 1, as the system attempts to reattain equilibrium following the electrolytic depletion of Red] in phase 1. 

The first step in the process is to relate heat flow to a temperature gradient, just as a diffusive flux can be related to a concentration gradient. The fundamental law of heat conduction was proposed by Jean Fourier in 1807 and relates the heat flux (q) to the temperature gradient ... 

I. The direction of diffusion or movement by solutes is always from a region of higher concentration to one of lower concentration. In addition, the diffusive flux, /a, of a solute A across a plane at x is equal to -D dcJ dx) where D is the diffusion coefficient and dcj dx is the concentration gradient of A at x. 2. The increase of the concentration of A with time, dcjdt, is equal to D d CA.lQx ) where d cjdx is the change in the concentration gradient. See also Diffusion Transport Processes... 

For ease of solution, it is assumed that the spherical shape of the pellet is maintained throughout reaction and that the densities of the solid product and solid reactant are equal. Adopting the pseudo-steady state hypothesis implies that the intrinsic chemical reaction rate is very much greater than diffusional processes in the product layer and consequently the reaction is confined to a gradually receding interface between reactant core and product ash. Under these circumstances, the problem can be formulated in terms of pseudo-steady state diffusion through the product layer. The conservation equation for this zone will simply reflect that (in the pseudo-steady state) there will be no net change in diffusive flux so... 

Recall the distinction between advective and diffusive transport, which we made in Section 18.1 while traveling in the dining car through the Swiss Alps. We then introduced Fick s first law to describe the mass flux per unit area and time by diffusion or by any other random process (Eq. 18-6). Rewritten in terms of partial derivatives, the diffusive flux along the x-axis is ... 

The sorption of a penetrant by a class (a) membrane can be treated most simply, if the processes of sorption in the polymer matrix and into each kind of specific sorption site within it can be treated as mutually independent and hence additive. Under these conditions, each sorbed species may similarly be expected to contribute additively to the total diffusion flux. Hence, if there are N — 1 kinds of specific sorption sites, the overall sorption and permeability or diffusion coefficients can be written as... 

Since the fraction of empty sites in a zeolite channel determines the correlation factor (Section 5.2.2), as is well known from single-file diffusion in the pores of a membrane, the strong dependence of the diffusion coefficients on concentration can be understood. This is why a simple Nernst-Planck type coupling of the diffusive fluxes (see, for example, [H, Schmalzried (1981)]) is also not adequate. Therefore, we should not expect that sorption and desorption are symmetric processes having identical kinetics. Surveys on zeolite kinetics can be found in [A. Dyer (1988) J. Karger, D.M. Ruthven (1992)]. 

By definition chirality involves a preferred sense of rotation in a three-dimensional space. Therefore, it can only be affected by a modification of the nonscalar fields appearing in the rate equations. For a reaction-diffusion system [equations (1)] these fields are descriptive of a vector irreversible process, namely, the diffusion flux J of constituent k in the medium. According to irreversible thermodynamics, the driving force conjugate to diffusion is... 

Here, the diffusion in the direction of flow is neglected as the transport process is dominated by convection. This may not be a valid assumption for relatively short channels. The normal diffusion flux of the specie, g d, is calculated using Equation (5.26) where [Pg.139]

N diffuses into the structural pores of clinoptilolite 10 to 10 times faster than does CH4. Thus internal surfaces are kinetically selective for adsorption. Some clino samples are more effective at N2/CH4 separation than others and this property was correlated with the zeolite surface cation population. An incompletely exchanged clino containing doubly charged cations appears to be the most selective for N2. Using a computer-controlled pressure swing adsorption apparatus, several process variables were studied in multiple cycle experiments. These included feed composition and rates, and adsorber temperature, pressure and regeneration conditions. N2 diffusive flux reverses after about 60 seconds, but CH4 adsorption continues. This causes a decay in the observed N2/CH4 separation. Therefore, optimum process conditions include rapid adsorber pressurization and short adsorption/desorp-tion/regeneration cycles. 

Seston-S deposition probably is a more important process than dissimilatory reduction in lakes with low [SO42 ]. As lakewater sulfate concentrations increase, seston deposition reaches a plateau limited by the overall primary production rate and the maximum algal S content, but diffusive fluxes continue to increase in direct proportion to [SO42 ]. Thus, in highly acidic lakes (pH 3 5 [SOjt2 J > 100 peq/L), such as McCloud Lake, Florida and Lake 223, Ontario, dissimilatory sulfate reduction probably is the major sulfate sink. Nriagu and Soon (131 concluded that endproducts of dissimilatory reduction and elevated sediment S content would not be observed below S mg/L (240 / eq/L), but we see clear evidence of dissimilatory reduction in Little Rock Lake at concentrations of approximately SO /teq/L. 

In-lake processes remove approximately half of the sulfate inputs from the water column of Little Rock Lake. Two processes, seston deposition and dissimilatory reduction, are responsible for sulfate retention. For the preacidified lake, seston deposition probably is the dominant sink, accounting for 70% of net retention. Preliminary data and theoretical considerations suggest that the diffusive flux of sulfate to sediments will increase during experimental acidification, and we believe that dissimilatoty reduction is the dominant sulfate sink in lakes with elevated sulfate concentrations. 

The process, which involves diffusion, is initiated by a temperature rise. Under these conditions, a diffusion flux (/diff) will emerge symmetrically from both sides of the sheet and the particle number density (n) will vary with time according to Fick s law ... 

We note first that immediately following the injection of a sample at the head of the channel, the flow of carrier is stopped briefly to allow time for the sample particles to accumulate near the appropriate wall. As the particles concentrate near the wall, the growing concentration gradient leads to a diffusive flux which counteracts the influx of particles. Because channel thickness is small (approximately 0.25 mm), these two mass transport processes quickly balance one another, leading to an equilibrium distribution near the accumulation wall. This distribution assumes the exponential form... 

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