Diffusional gradient

The dilemma is solved by taking into account the fact that the lack of an equal supply rate for cations and anions carried toward the electrodes by the electric current will create a concentration gradient near the interface for the slower ions, and this concentration gradient will speed up the motion of the slower ions to compensate for their poorer performance. It is this diffusional component that makes Faraday s laws come true. The diffusional gradient pitches in to help the slower ions to the electrode at the same rate as the faster ones. 

By "inert it means that the membrane is a separator but not a catalyst. Many membrane reactor modeling studies consider only those cases where the membrane is catalydcally inert and the catalyst is packed most often in the tubular (feed) region but sometimes in the annular (permeate) region. When it is assumed that no reaction takes place in the membrane or membrane/support matrix, the governing equations for the membrane/support matrix are usually eliminated. The overall eff ect of membrane permeation can be accounted for by a permeation term which appears in the macroscopic balance equations for both the feed and permeate sides. Thus, the diffusional gradient term... 

Transport Criteria in PBRs In laboratory catalytic reactors, basic problems are related to scaling down in order to eliminate all diffusional gradients so that the reactor performance reflects chemical phenomena only [24, 25]. Evaluation of catalyst performance, kinetic modeling, and hence reactor scale-up depend on data that show the steady-state chemical activity and selectivity correctly. The criteria to be satisfied for achieving this goal are defined both at the reactor scale (macroscale) and at the catalyst particle scale (microscale). External and internal transport effects existing around and within catalyst particles distort intrinsic chemical data, and catalyst evaluation based on such data can mislead the decision to be made on an industrial catalyst or generate irrelevant data and felse rate equations in a kinetic study. The elimination of microscale transport effects from experiments on intrinsic kinetics is discussed in detail in Sections 2.3 and 2.4 of this chapter. 

At the anode, concentration losses are caused by either insufficient supply of reduced species toward the electrode, or limited discharge of oxidised species from the electrode surface. This will increase the ratio between the oxidised and the reduced species at the electrode surface, which can produce an increased electrode potential (as demonstrated by the OCP ode curve in Figure 2.7c). At the cathode the reverse may occur, resulting in a drop in cathode potential (OCP(,3thode curve in Figure 2.7c). Moreover, diffusional gradients and thus mass transport limitations in the bulk liquid may occur in poorly mixed systems, which can also limit the substrate flux to the biofilm. 

One procedure makes use of a box on whose silk screen bottom powdered desiccant has been placed, usually lithium chloride. The box is positioned 1-2 mm above the surface, and the rate of gain in weight is measured for the film-free and the film-covered surface. The rate of water uptake is reported as u = m/fA, or in g/sec cm. This is taken to be proportional to - Cd)/R, where Ch, and Cd are the concentrations of water vapor in equilibrium with water and with the desiccant, respectively, and R is the diffusional resistance across the gap between the surface and the screen. Qualitatively, R can be regarded as actually being the sum of a series of resistances corresponding to the various diffusion gradients present ... 

Membra.ne Diffusiona.1 Systems. Membrane diffusional systems are not as simple to formulate as matrix systems, but they offer much more precisely controlled and uniform dmg release. In membrane-controlled dmg deUvery, the dmg reservoir is intimately surrounded by a polymeric membrane that controls the dmg release rate. Dmg release is governed by the thermodynamic energy derived from the concentration gradient between the saturated dmg solution in the system s reservoir and the lower concentration in the receptor. The dmg moves toward the lower concentration at a nearly constant rate determined by the concentration gradient and diffusivity in the membrane (33). 

For catalytic investigations, the rotating basket or fixed basket with internal recirciilation are the standard devices nowadays, usually more convenient and less expensive than equipment with external recirculation. In the fixed basket type, an internal recirculation rate of 10 to 15 or so times the feed rate effectively eliminates external diffusional resistance, and temperature gradients. A unit holding 50 cm (3.05 in ) of catalyst can operate up to 800 K (1440 R) and 50 bar (725 psi). 

The diffusion coefficients of the constituent ions in ionic liquids have most commonly been measured either by electrochemical or by NMR methods. These two methods in fact measure slightly different diffusional properties. The electrochemical methods measure the diffusion coefficient of an ion in the presence of a concentration gradient (Pick diffusion) [59], while the NMR methods measure the diffusion coefficient of an ion in the absence of any concentration gradients (self-diffusion) [60]. Fortunately, under most circumstances these two types of diffusion coefficients are roughly equivalent. 

Diffusion as referred to here is molecular diffusion in interstitial water. During early diagenesis the chemical transformation in a sediment depends on the reactivity and concentration of the components taking part in the reaction. Chemical transformations deplete the original concentration of these compounds, thereby setting up a gradient in the interstitial water. This gradient drives molecular diffusion. Diffusional transport and the kinetics of the transformation reactions determine the net effectiveness of the chemical reaction. 

The relationship between the diffusional flux, i.e., the molar flow rate per unit area, and concentration gradient was first postulated by Pick [116], based upon analogy to heat conduction Fourier [121] and electrical conduction (Ohm), and later extended using a number of different approaches, including irreversible thermodynamics [92] and kinetic theory [162], Pick s law states that the diffusion flux is proportional to the concentration gradient through... 

It must be pointed out that in a diffusion layer where the ions are transported not only by migration but also by diffusion, the effective transport numbers t of the ions (the ratios between partial currents ij and total current t) will differ from the parameter tj [defined by Eq. (1.13)], which is the transport number of ion j in the bulk electrolyte, where concentration gradients and diffusional transport of substances are absent. In fact, in our case the effective transport number of the reacting ions in the diffusion layer is unity and that of the nonreacting ions is zero. 

Thus, the potential difference in electrolytes during current flow is determined by two components an ohmic component (po m proportional to current density and a diffusional component q>, which depends on the concentration gradients. The latter arises only when the Dj values of the individual ions differ appreciably when they are all identical, is zero. The existence of the second component is a typical feature of electrochemical systems with ionic concentration gradients. This component can exist even at zero current when concentration gradients are maintained artificially. When a current flows in the electrolyte, this component may produce an apparent departure from Ohm s law. 

As the diffusional field strength depends on the coordinate jc in the diffusion layer, the diffusion flux density (in contrast to the total flux density) is no longer constant and the concentration gradients dCjIdx will also change with the coordinate x. 

It follows that convection of the hqnid has a twofold influence It levels the concentrations in the bnlk liquid, and it influences the diffusional transport by governing the diffusion-layer thickness. Shght convection is sufficient for the first effect, but the second effect is related in a qnantitative way to the convective flow velocity The higher this velocity is, the thinner will be the diffusion layer and the larger the concentration gradients and diffusional fluxes. 

Diffusion of ions can be observed in multicomponent systems where concentration gradients can arise. In individnal melts, self-diffnsion of ions can be studied with the aid of radiotracers. Whereas the mobilities of ions are lower in melts, the diffusion coefficients are of the same order of magnitude as in aqueous solutions (i.e., about 10 cmVs). Thus, for melts the Nemst relation (4.6) is not applicable. This can be explained in terms of an appreciable contribntion of ion pairs to diffusional transport since these pairs are nncharged, they do not carry cnrrent, so that values of ionic mobility calculated from diffusion coefficients will be high. 

In diffusional mass transfer, the transfer is always in the direction of decreasing concentration and is proportional to the magnitude of the concentration gradient the constant of proportionality being the diffusion coefficient for the system. 

DOSY is a technique that may prove successful in the determination of additives in mixtures [279]. Using different field gradients it is possible to distinguish components in a mixture on the basis of their diffusion coefficients. Morris and Johnson [271] have developed diffusion-ordered 2D NMR experiments for the analysis of mixtures. PFG-NMR can thus be used to identify those components in a mixture that have similar (or overlapping) chemical shifts but different diffusional properties. Multivariate curve resolution (MCR) analysis of DOSY data allows generation of pure spectra of the individual components for identification. The pure spin-echo diffusion decays that are obtained for the individual components may be used to determine the diffusion coefficient/distribution [281]. Mixtures of molecules of very similar sizes can readily be analysed by DOSY. Diffusion-ordered spectroscopy [273,282], which does not require prior separation, is a viable competitor for techniques such as HPLC-NMR that are based on chemical separation. 

The percutaneous absorption picture can be qualitatively clarified by considering Fig. 3, where the schematic skin cross section is placed side by side with a simple model for percutaneous absorption patterned after an electrical circuit. In the case of absorption across a membrane, the current or flux is in terms of matter or molecules rather than electrons, and the driving force is a concentration gradient (technically, a chemical potential gradient) rather than a voltage drop [38]. Each layer of a membrane acts as a diffusional resistor. The resistance of a layer is proportional to its thickness (h), inversely proportional to the diffusive mobility of a substance within it as reflected in a... 

Equation (9) is Fick s second law of diffusion, derived on the assumption that D is constant. Fick s second law essentially states that the rate of change in concentration in a volume within the diffusional field is proportional to the rate of change in the spatial concentration gradient at that point in the field, the proportionality constant being the diffusion coefficient. 

The self-diffusion coefficient is determined by measuring the diffusion rate of the labeled molecules in systems of uniform chemical composition. This is a true measure of the diffusional mobility of the subject species and is not complicated by bulk flow. It should be pointed out that this quantity differs from the intrinsic diffusion coefficient in that a chemical potential gradient exists in systems where diffusion takes place. It can be shown that the self-diffusion coefficient, Di, is related to the intrinsic diffusion coefficient, Df, by... 

The numerator of the right side of this equation is equal to the chemical reaction rate that would prevail if there were no diffusional limitations on the reaction rate. In this situation, the reactant concentration is uniform throughout the pore and equal to its value at the pore mouth. The denominator may be regarded as the product of a hypothetical diffusive flux and a cross-sectional area for flow. The hypothetical flux corresponds to the case where there is a linear concentration gradient over the pore length equal to C0/L. The Thiele modulus is thus characteristic of the ratio of an intrinsic reaction rate in the absence of mass transfer limitations to the rate of diffusion into the pore under specified conditions. 

So far, we have assumed that the particle is isothermal and have focused only on the diffusional characteristics and concentration gradient within the particle, and their effect on 17. We now consider the additional possibility of a temperature gradient arising from the thermal characteristics of the particle and the reaction, and its effect on tj. 

Equation 1.70 shows that the molar diffusional flux of component A in the y-direction is proportional to the concentration gradient of that component. The constant of proportionality is the molecular diffusivity 2. Similarly, equation 1.69 shows that the heat flux is proportional to the gradient of the quantity pCpT, which represents the. concentration of thermal energy. The constant of proportionality klpCp, which is often denoted by a, is the thermal diffusivity and this, like 2, has the units m2/s. 

Figure 8.11 Evidence of cross-diffusional effects. The homogeneous distribution of species 2 (dashed line, top) is perturbed by a coexisting gradient of species 1 (bottom).

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