Diffusion and concentration

Combined Pore and Solid Diffusion In porous adsorbents and ion-exchange resins, intraparticle transport can occur with pore and solid diffusion in parallel. The dominant transport process is the faster one, and this depends on the relative diffusivities and concentrations in the pore fluid and in the adsorbed phase. Often, equilibrium between the pore fluid and the solid phase can be assumed to exist locally at each point within a particle. In this case, the mass-transfer flux is expressed by ... 

Figure 3-33 Comparison of (a) Dout during dehydration (outward H2O diffusion) and (b) Din during hydration (inward H2O diffusion) and concentration-dependent D.

The diffusion equation is dc/dt = DBV2c, where DB and c are, respectively, the diffusivity and concentration of the B atoms in the a phase. The initial condition... 

Figure 1 shows the schematic of a tubular reactor, of radius a and length L, where a — a/Lis the aspect ratio. Clearly, ifa>S> 1, or a <3C 1, a physical length scale separation exists in the reactor. This length scale separation could also be interpreted in terms of time scales. For example, a 1 implies that the time scale for radial diffusion is much smaller than that of either convection and axial diffusion, and concentration gradients in the transverse direction are small compared to that in the axial direction. 

Brimhall, J.L., E.P. Simonen, and R.H. Jones, Data Base on Permeation, Diffusion, and Concentration of Hydrogen Isotopes in Fusion Reactor Materials, Fusion Reactor Materials Semiannual Progress Report, DOE/ER-0313/16, 1994. 

The mode of transport through a membrane may be passive, active, or facilitated type. In passive transport, the membrane acts as a barrier and permeation of the components is determined by their diffusivity and concentration in the membrane or just by their size. In facilitated transport along with the chemical potential gradient, the mass transport is coupled to specific carrier components in the membrane. In active transport driving force for transport is achieved by a chemical reaction in the membrane phase. 

Greenlaw et al. [38] proposed another relationship between diffusivity and concentration ... 

An alternative method to take surface diffusion into account consists in lumping pore diffusion and concentration-dependent surface diffusion together, thus creating an apparent effective diffusion coefficient, which is concentration dependent. This approach was used by Ma et al [53], by Pigtkowski et al. [28] and by Zhou et al. [10]. This method is also an approximation, but it is still an improvement over the simpler HSDM model. 

The basic concept of diffusion refers to the net transport of material within a single phase in the absence of mixing (by mechanical means or by convection). Both experiment and theory have shown that diffusion can result from pressure gradients (pressure diffusion), temperature gradients (thermal diffusion), external force fields (forced diffusion), and concentration gradients. Only the last type is considered in this book that is, the discussion is limited to diffusion caused by the concentration difference between two points in a stagnant solution. This process, called molecular diffusion, is described by Pick s laws. His first law relates the flux of a chemical to the concentration gradient ... 

Its diffusivity and concentration must be large enough to provide a sufficiently large demulsifier flux toward the surfaces and thus eliminate the gradients of the interfacial tension. 

In the case of a solid catalyst operating in a liquid phase reaction system the problems of diffusion and concentration gradients can be particularly severe. Substrate diffusion can be further broken down into two steps, external diffusion and internal diffusion. The former is controlled by the flow of substrate molecules through the layer of molecules surrounding catalyst particles and is proportional to the concentration gradient in the bulk liquid, i.e. the difference in the concentrations of the substrate in the bulk medium and at the catalyst surface. The thickness of the external layer in a liquid medium is dependent on the flowing fluid and on the agitation within the reaction system typically it is 0.1-0.01 mm thick. Internal diffusion of substrate molecules is a complex process determined not only by the resistance to flow due to the... 

The permeabilities of different components in a membrane depend on the mechanism by which the components are transported. For example, in homogeneous polymer membranes, the various chemical species are transported under a concentration or pressure gradient by diffusion. The permeability of these membranes is determined by the diffusivities and concentrations of the various components in the membrane matrix and the transport rates are, in general, relatively slow. In porous membrane structures, however, mass is transported under the driving force of a hydrostatic pressure difference via viscous flow and, in gen-... 

Figure 9 Concentration dependence of the self-diffusion coefficient of /j-hexane in zeolite NaX with mean crystallite diameters of 55 p-m (O), 20 xm([ 1), 15 xm ( 0 ), and 4 p.m (A) at 293 K. The proton resonance frequencies were 60 MHz (open symbols) and 16.6 MHz (full symbols), corresponding to external magnetic fields of 1.41 and 0.39 T, respectively. The error bars indicate the uncertainty in the diffusivities and concentrations. (From Ref. 108.)...

Calculation of Diffusion and Concentration of Oxygen in Ionic Liquids... 

The diffusion and concentration of oxygen are important parameters from the oxygen reduction reaction point of view. Thus, the calculation of these parameters in ionic liquids can be obtained using different methodologies reported in the literature. 

After the specimen is loaded in tension at intermediate stress levels, an incubation period is required to initiate a crack. During this time, lattice-contained or weakly-trapped hydrogen diffuses ahead of the notches and local stress raisers, so as to equilibrate under the influence of any triaxial stress created there-this equilibration involves hydrogen redistribution associated with the equilibration of its chemical potential, under the combined influences of all factors that affect it. The first cracks will be initiated in this region after a critical hydrogen concentration has accumulated. The kinetics of the crack-initiation process are therefore dependent on the temperature and the stress gradient as they relate to the diffusion and concentration of hydrogen at the site of maximum triaxial stress in the steel. 

Hydrogels such as the Q[6]-mediated alginate hydrogel beads discussed in Section 2.2.4 release their drug loads through a process of diffusion and concentration gradients unless some other mechanisms are built into the gel to modify release. This may be a direction for the future for certain drug applications. 

The rate of the chemical reaction is slow compared to internal and external diffusion, and concentration gradients both of the solid and gaseous reactant are negligible. 

The ionic or electrical conductivity included in this section is due to ionic motion of species i within metal oxide conpounds. These oxides may consist of one or more phases making up an oxide scale, which in turn, is the corrosion product to be analyzed with regard to its protectiveness, conductivity, mobility, diffusivity, and concentration of the species i. 

Enderlein J, Gregor I, Patra D, Dertinger T, Kaupp UB (2005) Perframance of fluorescence correlation spectroscopy for measuring diffusion and concentration. ChemPhysChem 6 2324-2336... 

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