Transport by diffusion

The quantity k is related to the intensity of the turbulent fluctuations in the three directions, k = 0.5 u u. Equation 41 is derived from the Navier-Stokes equations and relates the rate of change of k to the advective transport by the mean motion, turbulent transport by diffusion, generation by interaction of turbulent stresses and mean velocity gradients, and destmction by the dissipation S. One-equation models retain an algebraic length scale, which is dependent only on local parameters. The Kohnogorov-Prandtl model (21) is a one-dimensional model in which the eddy viscosity is given by... 

Nonporous Dense Membranes. Nonporous, dense membranes consist of a dense film through which permeants are transported by diffusion under the driving force of a pressure, concentration, or electrical potential gradient. The separation of various components of a solution is related directiy to their relative transport rate within the membrane, which is determined by their diffusivity and solubiUty ia the membrane material. An important property of nonporous, dense membranes is that even permeants of similar size may be separated when their concentration ia the membrane material (ie, their solubiUty) differs significantly. Most gas separation, pervaporation, and reverse osmosis membranes use dense membranes to perform the separation. However, these membranes usually have an asymmetric stmcture to improve the flux. 

Uncharged reaction components are transported by diffusion and convection, even though their migration fluxes are zero. The total flux density Jj of species j is the algebraic (vector) sum of densities of all flux types, and the overall equation for mass balance must be written not as Eq. (4.1) but as... 

Galvanic cells that include at least one electrolyte-electrolyte interface (which may be an interface with a membrane) across which ions can be transported by diffusion are called cells with transference. For the electrolyte-electrolyte interfaces considered in earlier sections, cells with transference can be formulated, for example, as... 

U throughout the water colutim. The length of gray arrows represents the size of the fluxes illustrating that °Th is rapidly scavenged everywhere, while Pa can be advected or transported by diffusion (not shown) from areas of low to high productivity. Sedimentary Pa/ °Th is therefore a fimction of both the productivity and the net lateral transport of Pa by ocean circulation. 

Radon from the soil enters into buildings by convective flow of soil gas. Transport by diffusion is normally insignificant. In houses with very high radon concentration, diffusion need not be considered because it can only provide an insignificant fraction of the source strength. There are three conditions necessary for infiltration of soil gas containing radon into the building from the soil ... 

I consider a system in which organic matter is oxidized at a steady rate that is a specified function of depth in uniform calcium carbonate sediments. The oxidation of organic matter increases the total dissolved carbon in the pore water of the sediment. The resultant increase in acidity causes the dissolution of calcium carbonate and a consequent increase in alkalinity as well as another increase in total dissolved carbon. The total dissolved carbon and alkalinity are transported by diffusion between different depths in the sediment. 

More than one point defect species may be present in a crystal at any temperature, and the amount of matter transported by diffusion will depend upon the number of each defect type present. In general, therefore, the overall apparent diffusion coefficient, D, will be the sum of the individual contributions, for example ... 

Holub, K. (1966). Surface reaction of an adsorbed substance transported by diffusion to a spherical electrode, Collect. Czech. Chem. Commun., 31, 1655-1665. 

Oxygen, substrate and biomass are all transported by diffusion within the liquid phase contained in the aggregate. The modelling of this process is achieved via the use of a finite differencing technique. In this, the spherical aggregate is divided into a number of shells, as seen in Fig. 1. 

It is assumed i) that the concentration c remains constant and ii) that transport by diffusion is rate controlling, i.e., the adsorbate arriving at the interface is adsorbed fast (intrinsic adsorption). This intrinsic adsorption, i.e., the transfer from the solution to the adsorption layer is not rate determining or in other words, the concentration of the adsorbate at the interface is zero iii) furthermore, the radius of the adsorbing particle is relatively large (no spherical diffusion). 

Theoretical dependence of filter efficiency of a single collector (proportional to the rate at which particle contacts occur between particles and the filter grain by mass transport) on particle diameter. For particles of small diameters transport by diffusion increases with decreasing size. Contact opportunities of the larger particles with the filter grain are due to interception and sedimentation they increase with increasing size. 

Mass transport by diffusion can never be totally eliminated if there are differences in concentration throughout the solution (e.g. as caused by current flow), but convection is such an efficient form of mass transport, when compared with either migration or diffusion, that it is safe to assume that diffusion is quite negligible in comparison. 

As already shown by Wiese et al. [17] mass transport rates in biphasic catalysis can be dramatically influenced by hydrodynamics in a tube reactor with Sulzer packings. Above all, the volume rate of the catalyst phase in which the substrates are transported by diffusion plays a decisive role in accelerating the mass transport rate. This effect was also investigated for citral hydrogenation in the loop reactor. Overall reaction rates and conversions as a function of the catalyst volume rate can be seen in Fig. 15. 

The effectiveness of the internal O2 transport by diffusion or convection depends on the physical resistance to movement and on the O2 demand. The physical resistance is a function of the cross-sectional area for transport, the tortuosity of the pore space, and the path length. The O2 demand is a function of rates of respiration in root tissues and rates of loss of O2 to the soil where it is consumed in chemical and microbial reactions. The O2 budget of the root therefore depends on the simultaneous operation of several linked processes and these have been analysed by mathematical modelling (reviewed by Armstrong... 

As follows from the hydrodynamic properties of systems involving phase boundaries (see e.g. [86a], chapter 2), the hydrodynamic, Prandtl or stagnant layer is formed during liquid movement along a boundary with a solid phase, i.e. also at the surface of an ISE with a solid or plastic membrane. The liquid velocity rapidly decreases in this layer as a result of viscosity forces. Very close to the interface, the liquid velocity decreases to such an extent that the material is virtually transported by diffusion alone in the Nernst layer (see fig. 4.13). It follows from the theory of diffusion transport toward a plane with characteristic length /, along which a liquid flows at velocity Vo, that the Nernst layer thickness, 5, is given approximately by the expression,... 

Water from the bulk of the melt is transported, by diffusion and forced convection, in a rate-controlling step to the electrode surface (this accounts for the proportionality of the limiting current to water concentration) at the electrode interface water... 

After the experiments, significant quantities of newly formed minerals were observed at the cold extremity of the tube, pointing to a fast material transport by diffusion from the hot to the cold end of the tube. The following spatial distribution of newly formed phases, reflecting the temperature profile, was observed in both runs (Fig. 8) quartz + K-feldspar + plagioclase + Mg-rich saponites (hot extremity) quartz + K-feldspar + plagioclase (middle of the tube) and alkaline or Ca-rich clays + quartz + plagioclase (cold extremity). The cation composition of the phyllosilicates was similar in both experiments. Some newly fonned quartz crystals... 

The tube-in-tube experiment is a very powerful method to determine the sequence of precipitation of secondary minerals as function of temperature for a chosen chemical system. Chemical reactions occur quickly (within 40 days) and the transport by diffusion of chemical elements is efficient. Similar crystallization sequences are observed in both experiments, suggesting that the transitions between the different mineral phases are not only controlled by the composition of the solution but also by temperature. The experimental design does not strictly correspond to the geometry encountered at the Soultz-sous-Forets site and therefore needs to... 

Electrochemical gas detection instruments have been developed which use a hydrated solid polymer electrolyte sensor cell to measure the concentration of specific gases, such as CO, in ambient air. These instruments are a spin-off of GE aerospace fuel cell technology. Since no liquid electrolyte is used, time-related problems associated with liquid electrolytes such as corrosion or containment are avoided. This paper describes the technical characteristics of the hydrated SPE cell as well as recent developments made to further improve the performance and extend the scope of applications. These recent advances include development of NO and NO2 sensor cells, and cells in which the air sample is transported by diffusion rather than a pump mechanism. 

Let us now consider coagulation of particles in the absence of any repulsive barrier. In addition, we assume that, although there are no interparticle forces that contribute to the transport of particles toward each other, there is sufficient attraction between the particles on contact for them to form a permanent bond. As early as 1917, Smoluchowski formulated the equations for the collision rate for particles transported by diffusion alone (Smoluchowski 1917), and we develop the same idea here. 

Table 223 Critical Distance Z,crit at Which the Influence of Diffusion and Advection Is Equal (Eq. 22-14) For L Lcnl, transport by diffusion is negligible compared to advective transport.

Exercise. Write the M-equation for the local distribution of charge carriers in the semiconductor of VI.9, assuming that they are transported by diffusion. 

The region around the electrode, through which transport occurs, is filled with solution and should usually be unimpeded by cell walls, other electrodes, etc. for a sufficient distance. How far is sufficient depends upon the mode of transport and on the duration of the experiment. For transport by diffusion alone, the requirements are very modest indeed, as indicated in Table 1. 

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