Diffusion-controlled model concentration profiles

Reaction rates of nonconservative chemicals in marine sediments can be estimated from porewater concentration profiles using a mathematical model similar to the onedimensional advection-diffusion model for the water column presented in Section 4.3.4. As with the water column, horizontal concentration gradients are assumed to be negligible as compared to the vertical gradients. In contrast to the water column, solute transport in the pore waters is controlled by molecular diffusion and advection, with the effects of turbulent mixing being negligible. 

Once the theoretical curves have been fitted (Figs. 2.10 and 2.11), it is possible to plot the concentration profiles of all the species included in the model and to determine the optimum thickness of the enzyme layer (Fig. 2.12). Because the Thiele modulus is the controlling parameter in the diffusion-reaction equation, it is obvious from (2.22) that the optimum thickness will depend on the other constants and functions included in the Thiele modulus. For this reason, the optimum thickness will vary from one enzyme and one kinetic scheme to another. 

FIG. 7-14 Concentration profiles with intraparticle diffusion control, = 70. ii absence of gas particle mass-transfer resistance. [From Wen, Noncatalytic Heterogeneous Solid-Fluid Reaction Models, Ind. Eng. Chem. 60(9) 34-54 (1968), Fig. 12.]... 

Laboratory column experiments were used to identify potential rate-controlling mechanisms that could affect transport of molybdate in a natural-gradient tracer test conducted at Cape Cod, Mass. Column-breakthrough curves for molybdate were simulated by using a one-dimensional solute-transport model modified to include four different rate mechanisms equilibrium sorption, rate-controlled sorption, and two side-pore diffusion models. The equilibrium sorption model failed to simulate the experimental data, which indicated the presence of a ratecontrolling mechanism. The rate-controlled sorption model simulated results from one column reasonably well, but could not be applied to five other columns that had different input concentrations of molybdate without changing the reaction-rate constant. One side-pore diffusion model was based on an average side-pore concentration of molybdate (mixed side-pore diffusion) the other on a concentration profile for the overall side-pore depth (profile side-pore diffusion). 

The concentration profile in the immobile-water phase is controlled by a diffusional-transport mechanism. The transfer rate from the immobile-water phase to the flowing-water phase is the diffusive flux, which depends on the concentration gradient in the immobile-water phase at the interface. Parameters V, 6, A, and Lg in the profile side-pore model are estimated from the shape of the breakthrough curve for a nonreactive tracer. Parameters Pbf and pbs are estimated from the shape of the breakthrough curve for a reactive solute. The effective molecular diffusivity Dm is estimated from values published in the literature. 

The first step necessary to release the drug is water penetration with formation of a gelified outer layer. For water-soluble cellulose derivatives, both water penetration and drug release are generally controlled by Fickian diffusion and are described by the so-called Higuchi s equations (see e.g. [14]). Recently, we have developed a model that can predict the solute (drug) and penetrant (water) concentration profiles with simultaneous swelling [76] ... 

Solution of Eq. 74 for the steady state gives the concentration profiles of photogenerated carriers in the nanostructured electrode. A priori separation of migration and diffusion is difficult, and most analytical models have been based on the assumption that diffusion is predominant. Therefore in order to simplify the analysis, the boundary conditions are chosen to be appropriate for diffusion controlled transport. Initially it is assumed that recombination is absent. With dn x,t)/dt and df E, X, t)/dt equal to zero, Eq. (74) simplifies to... 

This equation is an example of an equation taken from the literature and used to compare theory and experiment. It has been frequently used to confirm the potential range where an electrode reaction is diffusion controlled and also to estimate values of diffusion coefficients. It was stressed earlier than when taking an equation from the literature it is important to ensure that the experimental conditions are appropriate to the model used in developing the equation. In this case it would be essential to work (a) with a large excess of base electrolyte in the solution, (b) with a solution which is unstirred, and, ideally, it should be thermostatted, (c) only on a timescale less than 10s (note also that below 1ms, the data will be adversely affected by a charging current, see later), and (d) sufficient time must be left after each experiment for the concentration profile for O to be restored to the correct initial condition, i.e. Cq = Cq at all x, before a further experiment is attempted. 

This two step oxidation is necessary due to the thermodynamic incompatibility of CO and O2, which requires CO2 to be the oxidant. They set up the necessary flux equations for each region and derive concentration profiles and parabolic rate constants. As expected for gas-phase diffusion control, the temperature dependence is weak. Agreement between the model and measured rate constants is good. 

The above studies showed that, for this particular adhesive system, the rate of diffusion of water through the adhesive to the interface was the rate controlling step. Now, if the diffusion of water through the adhesive is Fickian in nature, then the concentration profile of water as a function of time into the joint may be calculated [6,43,44], and such information may then be readily used in life-prediction models, as discussed below. It should be noted that the values of the diffusion coefficient given in Fig. 12b, which are very typical for structural epoxy adhesives, lead to the conclusion that, at ambient temperatures, it will take at least a year or more for the adhesive layer in a joint, say about 20 mm x 10 mm in size (as often used in single-overlap shear joints), to reach its uniform, equilibrium concentration of water, although of course, depending upon the details of the adhesive system , even complete failure of the joint due to environmental attack may have occurred well before this time is reached. 

In the former study, which involves modelling the adsorption kinetics, allowance is made for the effect of variation of total pressure on concentration and temperature profiles within a spherical particle and thus simulate the conditions of pressure swing adsorption. The other study by Boon et al is concerned with the behaviour of porous oxide-based catalyst spheres prepared by the sol-gel method. Although they deal with very different systems and circumstances, these two papers both draw attention to the importance of macroporosity in diffusion control and mass transport. 

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