Steady-state process diffusion

A membrane of thickness, L, is covered by an infinitely thin, impermeable surface film with cracks (5,6). A symmetry unit of a single repeat in the lattice is shown in Figure 6A. The length of the symmetry unit is b and the length of a continuous portion of the impermeable surface coating is 2q (Figure 6B). Within the membrane, Laplace s equation describes the steady state diffusion process,... 

Fick s first law represents steady-state diffusion. The concentration profile (the concentration as a function of location) is assumed constant with respect to time. In general, however, concentration profiles do change with time. To describe these non-steady-state diffusion processes use is made of Fick s second law, which is derived from the first law by combining it with the continuity equation [dni/dt = -(Jin - /out) = — VJ,]... 

Mass transport in MREF is a combination of steady state and non-steady state diffusion processes. The mass transfer limited current density (i,) is related to the reactant concentration gradient (Cb-Cs) and to the diffusion layer thickness (8) by Nemst using the following equation ... 

This equation is correct for steady state diffusion process in clay). The solution of the equation (41) is... 

In this section and the next we develop criteria for intermediate phase formation and for the related phenomenon of spontaneous emulsification. First, we consider quasi-steady-state diffusion processes leading to intermediate phase formation some time after initial contact of the phases. In the next section, intermediate phase formation on initial contact is discussed. 

For the fundamentals of non-steady state diffusion processes, the reader is referred to the standard textbooks, such as those by Bird et al. (I960), Welty et al. (1984) and Coulson and Richardson (1990). 

Other Models for Mass Transfer. In contrast to the film theory, other approaches assume that transfer of material does not occur by steady-state diffusion. Rather there are large fluid motions which constantiy bring fresh masses of bulk material into direct contact with the interface. According to the penetration theory (33), diffusion proceeds from the interface into the particular element of fluid in contact with the interface. This is an unsteady state, transient process where the rate decreases with time. After a while, the element is replaced by a fresh one brought to the interface by the relative movements of gas and Uquid, and the process is repeated. In order to evaluate a constant average contact time T for the individual fluid elements is assumed (33). This leads to relations such as... 

The driving force for the transfer process was the enhanced solubility of Br2 in DCE, ca 40 times greater than that in aqueous solution. To probe the transfer processes, Br2 was recollected in the reverse step at the tip UME, by diffusion-limited reduction to Br . The transfer process was found to be controlled exclusively by diffusion in the aqueous phase, but by employing short switching times, tswitch down to 10 ms, it was possible to put a lower limit on the effective interfacial transfer rate constant of 0.5 cm s . Figure 25 shows typical forward and reverse transients from this set of experiments, presented as current (normalized with respect to the steady-state diffusion-limited current, i(oo), for the oxidation of Br ) versus the inverse square-root of time. 

The calculation becomes more difficult when the polarization resistance RP is relatively small so that diffusion of the oxidized and reduced forms to and from the electrode becomes important. Solution of the partial differential equation for linear diffusion (2.5.3) with the boundary condition D(dcReJdx) = —D(d0x/dx) = A/sin cot for a steady-state periodic process and a small deviation of the potential from equilibrium is... 

Mass transfer phenomena usually are very effective on distance scales much larger than the dimensions of the cell wall and the double layer dimensions. Thicknesses of steady-state diffusion layers1 in mildly stirred systems are of the order of 10 5 m. Thus, one may generally adopt a picture where the local interphasial properties define boundary conditions while the actual mass transfer processes take place on a much larger spatial scale. 

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... 

One case is where the ions are traveling to the electrode by a process of diffusion. Then the steady-state diffusion problem can be looked at from the diffusion-layer point of view (Section 7.9). The variation of concentration with distance can be approximated to a linear variation, and the linear concentration gradient can be considered to occur over an effective distance of 8, die diffusion-layer thickness. Then the diffusion current is given by (Section 7.9.10). 

It is evident that the square wave charge-potential curves corresponding to surface-bound molecules behave in a similar way to the normalized current-potential ones observed for a soluble solution reversible redox process in SWV when an ultramicroelectrode is used (i.e., when steady-state conditions are attained), providing the analogous role played by 2sw (surface-bound species) and (soluble solution species), and also 2f (Eq- (7.93)) and the steady-state diffusion-limited current (7 css), see Sect. 2.7. This analogy can be made because the normalized converted charge in a surface reversible electrode process is proportional to the difference between the initial surface concentration (I ) and that... 

In agreement with other results (12,106), the steady-state diffusion of methanol in SAPO-34 was found to be a non-activated process, as shown in Fig. 14. This result means that the steady-state diffusivity measured at low temperatures can be used directly in the Weisz-Prater criterion at reaction conditions, namely, high temperatures. 

What has been done so far is to consider steady-state diffusion in which neither the flux nor the concentration of diffusing particles in various regions changes with time. In other words, the whole transport process is time independent. What happens if a concentration gradient is suddenly produced in an electrolyte initially in a time-invariant equilibrium condition Diffusion starts of course, but it will not immediately reach a steady state that does not change with time. For example, the distance variation of concentration, which is zero at equilibrium, will not instantaneously hit the final steady-state pattern. How does the concentration vary with time ... 

Description. The model organism is a free-floating unicellular sphere with characteristics selected, where possible, to match those of a phytoplankton cell. The organism and its environment (Figure Ic) are divided into four concentric zones -the bulk solution, the diffusion layer, the containing membrane and the cell concents. We will assume that the species taken up by the cell is the free metal ion since most of the studies of the uptake of B-subgroup metals by organisms support this hypothesis Z . 5 steady-state transport processes are... 

Solution of the transport problem when the process is controlled by both the interfacial electron transfer and the steady-state or linear diffusion of reactants was derived by Samec [181, 182]. These results represent the basis for the kinetic analysis, e.g., in dc polarography or convolution and potential sweep voltammetry. Under the conditions of steady-state diffusion, Eq. (60) can be transformed into a dimensionless form [181],... 

A simple, quantitative, steady-state diffusion model (36) demonstrates the importance of physical processes in shaping the vertical distribution of phytoplankton. This model uses values of the eddy diffusion coefficient K from the theoretical model of James (35), which reproduces accurately the annual cycle of vertical temperature structure for this area of the Celtic Sea. The submodels for photosynthetic production, light, and grazing can be varied to any of the established models nutrient luxury or nutrient limitation of growth can be included. The model reproduces the main features of the UOR observations in the Celtic Sea and English Channel. 

In order to study the principles of spatial discretization, we consider the steady state diffusion of a property in a one dimensional domain as sketched in Fig 12.3. In Cartesian coordinates the process is governed by ... 

In order to develop a continuous separation process, Kataoka et al. [54] simulated permeation of metal ion in continuous countercurrent column. They developed the material balance equation considering back mixing only in the continuous phase and steady-state diffusion in the dispersed emulsion drops which is similar to the Hquid extraction situation. Bart et al. [55] also modeled the extraction of copper in a continuous countercurrent column. They considered only the continuous phase back mixing in the model and assumed that the reaction between copper ions and carrier is slow, so that the differential mass balance equation for external phase in their model is... 

Following [367, 368], let us consider steady-state diffusion to a particle in a laminar flow. We assume that on the surface of the particle and remote from it, the concentration is constant and equal to Cs and C, respectively. In the dimensionless variables (3.1.7), the mass transfer process in the continuous medium is described by the equation... 

The small slope of the stage II section of the crack-growth rate versus K curve is attributed to corrosion-related, diffusion-controlled processes in the crack. Steady-state diffusion mechanisms are required to account for the fact that the crack growth rate is essentially constant... 

We apply simple effective medium models in an attempt to understand the diffusion process in the complex pore network of a porous SiC sample. There is an analogy between the quantities involved in the electrostatics problem and the steady state diffusion problem for a uniform external diffusion flux impinging on a coated sphere. Kalnin etal. [17] provide the details of such a calculation for the Maxwell Garnett (MG) model [18]. The quantity involved in the averaging is the product of the diffusion constant and the porosity for each component of the composite medium. The effective medium approach does not take into account possible effects due to charges on the molecules and/or pore surfaces, details in the size and shape of the protein molecules, fouling (shown to be negligible in porous SiC), and potentially important features of the microstructure such as bottlenecks. 

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