Diffusion from well-stirred solution

These data were used to check the hypothesis that diffusion of the condensed molybdenum oxide within the clay loam particles is the ratecontrolling step. As described earlier in this chapter, Crank (7) has given a graphical solution for the integrated diffusion equation for a sphere in a well-stirred solution. From the experimental measurements on the molybdenum oxide uptake by the hemisphere, all terms in the... 

Fig. 15. Diffusion of dilute, aqueous acetaminophen into a long, swollen cylinder of 10x4 PNIPAAm gel at 25 °C. The diffusion coefficient is extracted from a nonlinear least squares curve fit of the exact solution for diffusion into a cylinder of infinite length immersed in a well-stirred solution of finite volume to the data [123, 149]...

Various techniques are available for determining the effective diffusivity of solute in gel (Itamunoala, 1988). One of the most reliable techniques is the thin-disk method which uses a diffusion cell with two compartments divided by a thin gel. Each compartment contains a well-stirred solution with different solute concentrations. Effective diffusivity can be calculated from the mass flux verses time measurement (Hannoun and Stephanopoulos, 1986). A few typical values of effective diffusivities are listed in Table 3.2. 

In the case of oxidation in well-stirred solutions, however, neither the reaction rate nor the induction period depend on oxygen pressure. When the solution is vigorously stirred the auto-oxidation of polymers is unaffected by oxygen diffusion. If the system is not agitated, the local concentrations of oxygen and of radicals, as well as the oxidation rate, are functions of the distance from the surface. 

Several assumptions are made to mathematically model the immobilized adsorbent. The small adsorbent particles are assumed to be distributed uniformly inside the hydrogel bead. The external mass transfer resistance due to the boundary layer is assumed to be negligible if the bulk solution is well stirred. This assumption is supported by the experimental observations of Tanaka et al. who studied diffusion of several substrates from well stirred batch solutions into Ca-alginate gel beads (4), However, the boundary conditions can be easily modified to incorporate external diffusion effects if needed. Furthermore product diffusion in both the hydrogel and the porous adsorbent is considered to follow Fickian laws and its diffusivity in each region is assumed to be constant. 

Semi-infinite linear diffusion is considered in the Randles model, and the capacitive current is separated from the faradaic current, which is justified only when different ions take part in the double-layer charging and the charge transfer processes (i.e., a supporting electrolyte is present at high concentrations). Finite diffusion conditions should be considered for well-stirred solutions when the diffusion takes place only within the diSusion layer, and also in the case of siuface films that have a finite thickness. However, the two cases are different, since in the previous... 

When a diffusing solute partitions or adsorbs onto a solid matrix, we can often use standard solutions for nonsorbing solids to follow the course of adsorption by suitably modifying one of the solution parameters. For the case of adsorption by spherical particles from a well-stirred solution of limited volume, for example, the parameter Soi n/ spheres io Figure 4.4 is replaced by Vsoi-n/f Vspheres/ where K is the partition coefficient or Henry s constant. Assume the following parameter values K = 10, Vgoi-n/V spheres = 10, D = 10 cm /s, R = 0.46 cm. Whaf is the fractional saturation of the adsorbent after 1 h ... 

Example 2.2-1 Membrane diffusion Derive the concentration profile and the flux for a single solute diffusing across a thin membrane. As in the preceding case of a film, the membrane separates two well-stirred solutions. Unlike the film, the membrane is chemically different from these solutions. 

In the example above, the solutions are assumed to be well stirred and mixed the aqueous resistance is negligible, and the membrane is the only transport barrier. However, in any real case, the solutions on both sides of the membrane become less and less stirred as they approach the surface of the membrane. The aqueous diffusion resistance, therefore, very often needs to be considered. For example, for very highly permeable drugs, the resistance to absorption from the gastrointestinal tract is mainly aqueous diffusion. In the section, we give a general solution to steady diffusion across a membrane with aqueous diffusion resistance [5],... 

This point can be appreciated more quantitatively after consideration of an important (but simple) model of transport-controlled adsorption kinetics, the film diffusion process.34 35 This process involves the movement of an adsorptive species from a bulk aqueous-solution phase through a quiescent boundary layer ( Nemst film ) to an adsorbent surface. The thickness of the boundary layer, 5, will be largest for adsorbents that adsorb water strongly and smallest for aqueous solution phases that are well stirred. If j is the rate at which an... 

The starting point for modeling permeation (migration) to the liquid is the second case (ii). This is because it represents the well-studied diffusion of a solute from a polymer of limited volume, VP, into a stirred solution of limited volume, VL. A suitable equation for all of these cases is Eq. (7-51), where cP0 = cPe. 

An enlarged view of the two velocity components near the electrode surface is shown in Fig. 16D, for the same numerical parameters. The physical meaning of the Nemst diffusion layer becomes clear in this form of presentation. Thus, the perpendicular velocity component inside the Nemst diffusion layer is very small, not exceeding 2% of its value far away from the surface. This is the justification for the assumption that inside this diffusion layer the solution is practically stagnant, even though the solution as a whole is well stirred by rotating the electrode. 

The aforesaid situations apply in special cases where diffusion through the material in chamber A is not important (A is well stirred) and where the dissolution rate of the drug particles in A is rapid. A more common situation arises when drug release is both a function of its concentration within a vehicle and its ability to diffusion through it. When placed into a release medium, the drug closest to the surface is released the fastest. Over a period of time, the drug must diffuse from further and further back within the bulk of the device, which progressively slows the release. Systems such as this can be described by solutions to Ficks s second law of diffusion (1). 

The solution of Pick s second law for the diffusion of solute from a limited and constant volume of well-stirred gas into a spherical solid has already been worked ouP for a sphere of radius a when the mole fraction of in the gas is the same as that in the surface of the solid. The solution is... 

A third, much more difficult, example of convection and diffusion occurs in the apparatus shown schematically in Fig. 2.5-3. The apparatus consists of two well-stirred reservoirs. The upper reservoir contains a dense solution, but the lower one is filled with less dense solvent. Because solution and solvent are miscible, solute diffuses from the upper reservoir into the lower one. 

Water containing 0.1-M benzoic acid flows at 0.1 cm/sec through a 1-cm-diameter rigid tube of cellulose acetate, the walls of which are permeable to small electrolytes. These walls are 0.01 cm thick solutes within the walls diffuse as through water. The tube is immersed in a large well-stirred water bath. Under these conditions, the flux of benzoic acid from the bulk to the walls can be described by the correlation in Problem 8.2. After 50 cm of tube, what fraction of a 0.1-M benzoic acid solution has been removed Remember that there is more than one resistance to mass transfer in this system. 

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