Profiles 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,]... 

Consider the idealized picture of a mass-transfer process based on the two-film theory as shown in Figure 7.27. The partial pressure profile in the gas film is linear, as called for by steady-state diffusion, but the concentration profile in the liquid film falls below a linear measure as a result of a first-order chemical reaction removing the absorbed gas. A normal mass balance over the differential segment dz (we may assume unit area normal to the direction of diffusion) produces a familiar result ... 

The condition of steady state diffusion means that concentrations and fluxes at every point on the x-axis are constant with time. It then follows from eq. (5-29) for the case of a constant diffusion coefficient that the concentration gradient is locally constant everywhere, i. e. independent of x. In other words, there will be a linear concentration profile. Examples of such a steady state diffusional process would be the diffusion of numerous gases through metal foils when constant but different partial gas pressures are maintained on either side of the foils. In the system palladium-hydrogen, because of the high diffusivity of hydrogen, direct use is made of the diffusion of the gas through the metal for purposes of gas purification. 

In the second model an average striation thickness 8 is assumed, and with this the combined process of diffusion and reaction can be calculated as a function of time. For this one needs a set of combined differential equations such as eqs. (5.28a,b) (though with different boundary conditions see eqs. (5A.la-c) in the Appendix). The resulting concentration profiles are of the shape shown in figure 5.2 the distance between the peaks is the striation thickness. In principle, this striation thickness is a physical parameter that can be measured directly. For this model, a micro-mixing time could be defined by putting the Fourier- number for non-steady state diffusion equal to 0.1 (see eq. (4.8)) ... 

The concentration profiles are shown in Figure 6. As time approaches infinity, the term involving the exponential vanishes and the diffusion process approaches the steady state Eqs. (87) and (88) are then reduced to the steady concentration profile, Eq. (38), and flux, Eq. (39). 

Here F is the Faraday constant C = concentration of dissolved O2, in air-saturated water C = 2.7 x 10-7 mol cm 3 (C will be appreciably less in relatively concentrated heated solutions) the diffusion coefficient D = 2 x 10-5 cm2/s t is the time (s) r is the radius (cm). Figure 16 shows various plots of zm(02) vs. log t for various values of the microdisk electrode radius r. For large values of r, the transport of O2 to the surface follows a linear type of profile for finite times in the absence of stirring. In the case of small values of r, however, steady-state type diffusion conditions apply at shorter times due to the nonplanar nature of the diffusion process involved. Thus, the partial current density for O2 reduction in electroless deposition will tend to be more governed by kinetic factors at small features, while it will tend to be determined by the diffusion layer thickness in the case of large features. 

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. 

It is useful to point out here that we frequently encounter partial steady-states. An important example is the case where the diffusion process is much faster than a surface process, and thus a quasi-steady-state is reached for the diffusion concentration profile at each changing concentration of the surface. This distinction between different timescales of the processes can lead to a significant simplification of complex problems, see end of Section 4.3 or Chapter 4 in this volume. 

The last issue that remains to be addressed is whether the MBL results are sensitive to the characteristic diffusion distance L one assumes to fix the outer boundary of the domain of analysis. In the calculations so far, we took the size L of the MBL domain to be equal to the size h - a of the uncracked ligament in the pipeline. To investigate the effect of the size L on the steady state concentration profiles, in particular within the fracture process zone, we performed additional transient hydrogen transport calculations using the MBL approach with L = 8(/i — a) = 60.96 mm under the same stress intensity factor Kf =34.12 MPa /m and normalized T-stress T /steady state distributions of the NILS concentration ahead of the crack tip are plotted in Fig. 8 for the two boundary conditions, i.e. / = 0 and C, =0 on the outer boundary. The concentration profiles for the zero flux boundary condition are identical for both domain sizes. For the zero concentration boundary condition CL = 0 on the outer boundary, although the concentration profiles for the two domain sizes L = h - a and L = 8(/i - a) differ substantially away from the crack tip. they are very close in the region near the crack tip, and notably their maxima differ by less than... 

As we saw with the steady-state water-column application of the one-dimensional advection-diffusion-reaction equation (Eq. 4.14), the basic shapes of the vertical concentration profiles can be predicted from the relative rates of the chemical and physical processes. Figure 4.21 provided examples of profiles that exhibit curvatures whose shapes reflected differences in the direction and relative rates of these processes. Some generalized scenarios for sedimentary pore water profiles are presented in Figure 12.7 for the most commonly observed shapes. 

Another distinctive feature of strong tunnelling recombination could be seen after a step-like (sudden) increase (decrease) of temperature (or diffusion coefficient - see equation (4.2.20)) when the steady-state profile has already been reached. Such mobility stimulation leads to the prolonged transient stage from one steady-state y(r,T ) to another y(r,T2), corresponding to the diffusion coefficients D(T ) and >(72) respectively. This process is shown schematicaly in Fig. 4.2 by a broken curve. It should be stressed that if tunnelling recombination is not involved, there is no transient stage at all since the relevant steady state profile y(r) — 1 - R/r, equation (4.1.62), doesn t depend on >( ). 

The study of rotating disk electrode behavior provides a unique opportunity to develop a model that predicts the effect of diffusion and convection on the current. This is one of the few convective systems that have simple hydrodynamic equations that may be combined with the diffusion model developed herein to produce meaningful results. The effect of diffusion is modeled exactly as it has been done previously. The effect of convection is treated by integrating an approximate velocity equation to determine the extent of convective flow during a given At interval. Matter, then, is simply transferred from volume element to volume element in accord with this result to simulate convection. The whole process repeated results in a steady-state concentration profile and a steady-state representation of the current (the Levich equation). 

In the FFF a field or gradient is applied in a direction, perpendicular to the axis of a narrow flow channel. At the same time a solvent is forced steadily through the channel forming a cross-sectional flow profile of parabolic shape. When a polymer sample is injected into the channel, a steady state is soon reached in which the field induced motion and the opposed diffusion are exactly balanced. The continuous size-distribution of the polymer will migrate with a continuous spectrum of velocities and will emerge at the end of the flow channel with a continuous time distribution. When processed through a detector and its associated electronics, the time distribution becomes an elution (retention) spectrum. 

Diffusion time (diffusion time constant) — This parameter appears in numerous problems of - diffusion, diffusion-migration, or convective diffusion (- diffusion, subentry -> convective diffusion) of an electroactive species inside solution or a solid phase and means a characteristic time interval for the process to approach an equilibrium or a steady state after a perturbation, e.g., a stepwise change of the electrode potential. For onedimensional transport across a uniform layer of thickness L the diffusion time constant, iq, is of the order of L2/D (D, -> diffusion coefficient of the rate-determining species). For spherical diffusion (inside a spherical volume or in the solution to the surface of a spherical electrode) r spherical diffusion). The same expression is valid for hemispherical diffusion in a half-space (occupied by a solution or another conducting medium) to the surface of a disk electrode, R being the disk radius (-> diffusion, subentry -> hemispherical diffusion). For the relaxation of the concentration profile after an electrical perturbation (e.g., a potential step) Tj = L /D LD being - diffusion layer thickness in steady-state conditions. All these expressions can be derived from the qualitative estimate of the thickness of the nonstationary layer... 

The vividness of our world does not rely on processes that are characterized by linear force-flux relations, rather they rely on the nonlinearity of chemical processes. Let us recapitulate some results for proximity to equilibrium (see also Section VI.2.H.) In equilibrium the entropy production (n) is zero. Out of equilibrium, II = T<5 S/I8f > 0 according to the second law of thermodynamics. In a perturbed system the entropy production decreases while we reestablish equilibrium (II < 0), (Fig. 72). For the cases of interest, the entropy production can be written as a product of fluxes and corresponding forces (see Eq. 108). If some of the external forces are kept constant, equilibrium cannot be achieved, only a steady state occurs. In the linear regime this steady state corresponds to a minimum of entropy production (but nonzero). Again this steady state is stable, since any perturbation corresponds to a higher II-value (<5TI > 0) and n < 0.183 The linear concentration profile in a steady state of a diffusion experiment (described in previous sections) may serve as an example. With... 

Even if rate measurements in sediments are made using whole core incubations, e.g., when the inhibitor is a gas, it is still difficult to obtain a depth distribution of the rate (usually, an areal rate is obtained). A sophisticated measurement and model based system that avoids direct rate measurements has been used to overcome this problem. Microelectrodes, which have very high vertical resolution, are used to measure the fine scale distribution of oxygen and NOs" in freshwater sediments. By assuming that the observed vertical gradients represent a steady state condition, reaction-diffusion models can then be used to estimate the rates of nitrification, denitrification and aerobic respiration and to compute the location of the rate processes in relation to the chemical profiles (e.g., Binnerup et ai, 1992 Jensen et ai, 1994 Meyer et ai, 2001 Rysgaard et ai, 1994). Recent advances and details of the microelectrode approach can be found in the Chapter by Joye and Anderson (this volume). 

If the diffusion coefficient is constant, the moisture content profile through a material for the steady-state movement of moisture through it would be linear. However, drying is not a steady-state process. When the moisture content change occurs over almost the entire half thickness of the material, in other words when the size of the fully wet region is very small, the moisture content profiles can be shown to be parabolic during drying if the diffusion coefficient is constant. 

---