Diffusion steady-state conditions

Most theories of droplet combustion assume a spherical, symmetrical droplet surrounded by a spherical flame, for which the radii of the droplet and the flame are denoted by and respectively. The flame is supported by the fuel diffusing from the droplet surface and the oxidant from the outside. The heat produced in the combustion zone ensures evaporation of the droplet and consequently the fuel supply. Other assumptions that further restrict the model include (/) the rate of chemical reaction is much higher than the rate of diffusion and hence the reaction is completed in a flame front of infinitesimal thickness (2) the droplet is made up of pure Hquid fuel (J) the composition of the ambient atmosphere far away from the droplet is constant and does not depend on the combustion process (4) combustion occurs under steady-state conditions (5) the surface temperature of the droplet is close or equal to the boiling point of the Hquid and (6) the effects of radiation, thermodiffusion, and radial pressure changes are negligible. 

A more rigorous treatment takes into account the hydrodynamic characteristics of the flowing solution. Expressions for the limiting currents (under steady-state conditions) have been derived for various electrodes geometries by solving the three-dimensional convective diffusion equation ... 

If two vessels each containing completely mixed gas, one at temperature T, and the other at a temperature T2, are connected by a lagged non-conducting pipe in which there are no turbulent eddies (such as a capillary tube), then under steady state conditions, the rate of transfer of A by thermal diffusion and molecular diffusion must be equal and opposite, or. 

Under steady state conditions, the rate of diffusion inwards at the outside of the shell (r + dr) minus the rate of diffusion inwards at the inside of the shell (r) corresponds to the rate of reaction in the shell. In other words what came in and did not go out has reacted. The inward flux at position r is given by Pick s first law and hence the rate is the area of the shell A(r) = 4jtr multiplied by the flux ... 

In electrochemical systems with flat electrodes, all fluxes within the diffusion layers are always linear (one-dimensional) and the concentration gradient grad Cj can be written as dCfldx. For electrodes of different shape (e.g., cylindrical), linearity will be retained when thickness 5 is markedly smaller than the radius of surface curvature. When the flux is linear, the flux density under steady-state conditions must be constant along the entire path (throughout the layer of thickness 8). In this the concentration gradient is also constant within the limits of the layer diffusion layer 5 and can be described in terms of finite differences as dcjidx = Ac /8, where for reactants, Acj = Cyj - c j (diffusion from the bulk of the solution toward the electrode s surface), and for reaction products, Acj = Cg j— Cyj (diffusion in the opposite direction). Thus, the equation for the diffusion flux becomes... 

Regularly repeat i potential pulses will establish a steady state condition at the electrode surface where diffusion just replenishes the concentration of the compound... 

At steady-state conditions, the rate of supply of S by diffusion is balanced by the rate of consumption by chemical reaction, where assuming a first-order chemical reaction... 

Partial) dialysis in flow analysis. The sample solution flows along one side of the membrane, while the analyser solution passing (often in counter-current) on the other side takes up the diffused components from the sample. A dynamic equilibrium is reached (under steady-state conditions) in the leaving analyser solution, which is then analysed and from the result of which the analyte content can be derived via calibration with standard solutions treated in exactly the same way. This is a common procedure, e.g., in Technicon AutoAnalyzers, and has also been applied in haemoanalysis by Ammann et al.154 as described above. 

Convective diffusion to a growing sphere. In the polarographic method (see Section 5.5) a dropping mercury electrode is most often used. Transport to this electrode has the character of convective diffusion, which, however, does not proceed under steady-state conditions. Convection results from growth of the electrode, producing radial motion of the solution towards the electrode surface. It will be assumed that the thickness of the diffusion layer formed around the spherical surface is much smaller than the radius of the sphere (the drop is approximated as an ideal spherical surface). The spherical surface can then be replaced by a planar surface... 

Under steady-state conditions, the reaction rate is equal to the rate of diffusion of reactant through the poisoned region. The latter may be written as... 

It may be assumed that the accumulation of hydrogen within the pellet is negligible and that it may be treated as being in a quasi-steady-state condition. The finite difference form of Fick s first law may be used to determine the flow rate of hydrogen through the pellet. The diffusion constant appearing in this equation may be considered as an effective Knudsen diffusion coefficient. 

Equation (2.19), which concerns a situation without processes in the biofilm, can be extended to include transformation of a substrate, an electron donor (organic matter) or an electron acceptor, e.g., dissolved oxygen. If the reaction rate is limited by j ust one substrate and under steady state conditions, i.e., a fixed concentration profile, the differential equation for the combined transport and substrate utilization following Monod kinetics is shown in Equation (2.20) and is illustrated in Figure 2.8. Equation (2.20) expresses that under steady state conditions, the molecular diffusion determined by Fick s second law is equal to the bacterial uptake of the substrate. 

A large number of analytical solutions of these equations appear in the literature. Mostly, however, they deal only with first order reactions. All others require solution by numerical or other approximate means. In this book, solutions of two examples are carried along analytically part way in P7.02.06 and P7.02.07. Section 7.4 considers flow through an external film, while Section 7.5 deals with diffusion and reaction in catalyst pores under steady state conditions. 

Equation (14) also shows that for microorganisms with radii that are less than a few microns with a typical diffusion layer thickness > 10 pm, radial diffusion should predominate over linear diffusion [46], Under steady-state conditions, the area integrated cellular flux (mols-1), Q, for a small, spherical cell of surface = 4tt q, is given by ... 

Under steady-state conditions, the internalisation flux equals the rate of supply by diffusive transport and chemical reactions. As was shown earlier (cf. equations (12) and (13)), the maximum flux (rate) of solute internalisation by a microscopic cell under diffusion-limited conditions can be given by ... 

Alternatively, if the reactions at the surface are slow in comparison with diffusion or other reaction steps, the dissolution processes are controlled by the processes at the surface. In this case the concentrations of solutes adjacent to the surface will be the same as in the bulk solution. The dissolution kinetics follows a zero-order rate law if the steady state conditions at the surface prevail ... 

The intercellular route is considered to be the predominantly used pathway in most cases, especially when steady-state conditions in the stratum corneum are reached. In case of intercellular absorption, substance transport occurs in the bilayer-structured, continuous, intercellular lipid domain within the stratum corneum. Although this pathway is very tortuous and therefore much longer in distance than the overall thickness of the stratum corneum, the intercellular route is considered to yield much faster absorption due to the high diffusion coefficient of most drugs within the lipid bilayer. Resulting from the bilayer structure, the intercellular pathway provides hydrophilic and lipophilic regions, allowing more hydrophilic substances to use the hydrophilic and more lipophilic substances to use the lipophilic route. In addition, it is possible to influence this pathway by certain excipients in the formulation. 

The Gaussian expressions are not expected to be valid descriptions of turbulent diffusion close to the surface because of spatial inhomogeneities in the mean wind and the turbulence. To deal with diffusion in layers near the surface, recourse is generally had to the atmospheric diffusion equation, in which, as we have noted, the key problem is proper specification of the spatial dependence of the mean velocity and eddy difiusivities. Under steady-state conditions, turbulent diffusion in the direction of the mean wind is usually neglected (the slender-plume approximation), and if the wind direction coincides with the x axis, then = 0. Thus, it is necessary to specify only the lateral (Kyy) and vertical coefficients. It is generally assumed that horizontal homogeneity exists so that u, Kyy, and Ka are independent of y. Hence, Eq. (2.19) becomes... 

For a non-homogeneous membrane,both the diffusivity and salt distribution coefficient may vary as a function of position across the membrane. However, the steady-state conditions require that the molar salt flux and the total volume flux remain constant throughout the membrane. Therefore, the integrated expression,... 

Discussion. Fixed bed cracking reactors as well as commercial moving bed reactors operate under steady state or pseudo-steady state conditions ( ). Observed selectivity (eg., ratio of yield of branched to n-paraffin) in a steady state catalytic reactor is independent of space velocity (1, 17). The selectivity depends on intrinsic rate constants and diffusivities of the reacting species which depend on temperature. Thus, the selectivity observations reported here are applicable to commercial FCC units operating at space velocities different from that employed in this study. 

Transient and Steady-State Conditions From the landmine studies we readily conclude that the source term for these molecules has an initial spike, or increased rate, in the days or weeks after the mine is placed. This rate then decreases to some more or less constant level and may remain at that level for years. The initial spike comes from surface contamination, while the long-term rate is primarily from diffusion through the case and seals or leakage through imperfections or damage. The rates are clearly subject to environmental factors, principally temperature and soil wetness. Nevertheless, it seems clear that, at least in the case of landmines, there is a continuing flux of molecules that provide a potential for detection. 

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