Diffusion problems

Petera, J., Nassehi, V. and Pittman, J.F.T., 1989. Petrov-Galerkiii methods on isoparametric bilinear and biquadratic elements tested for a scalar convection-diffusion problem. Ini.. J. Numer. Meth. Heat Fluid Flow 3, 205-222,... 

Morton, K. W., 1996. Numerical Solution of Convection Diffusion Problems, Chapman Hall, London. 

In practice, 1—10 mol % of catalyst are used most of the time. Regeneration of the catalyst is often possible if deemed necessary. Some authors have advocated systems in which the catalyst is bound to a polymer matrix (triphase-catalysis). Here separation and generation of the catalyst is easy, but swelling, mixing, and diffusion problems are not always easy to solve. Furthermore, triphase-catalyst decomposition is a serious problem unless the active groups are crowns or poly(ethylene glycol)s. Commercial anion exchange resins are not useful as PT catalysts in many cases. 

Steady state pi oblems. In such problems the configuration of the system is to be determined. This solution does not change with time but continues indefinitely in the same pattern, hence the name steady state. Typical chemical engineering examples include steady temperature distributions in heat conduction, equilibrium in chemical reactions, and steady diffusion problems. 

Diffusion problems in one dimension lead to boundaiy value problems. The boundaiy conditions are applied at two different spatial locations at one side the concentration may be fixed and at the other side the flux may be fixed. Because the conditions are specified at two different locations, the problems are not initial value in character. It is not possible to begin at one position and integrate directly because at least one of the conditions is specified somewhere else and there are not enough conditions to begin the calculation. Thus, methods have been developed especially for boundary value problems. 

Example A reaction diffusion problem is solved with the finite difference method. 

The former usually involves process temperature or isolation. Sohds surface characteristics are important in that they control the extent to which an operation is diffusion-limited, i.e., diffusion into and out of the pores of a given sohds particle, not through the voids among separate particles. The size of the solids parti(des, the surface-to-mass ratio, is also important in the evaluation of surface characteristics and the diffusion problem. 

It is clear that the achievement of equilibrium is assisted by the maximum contact between the reactant and the Uansporting gas, but the diffusion problem is complex, especially in a temperature gradient, when a process known as thermal diffusion occurs. The ordinaty concenuation-dependent diffusion process occurs across dre direction of gas flow, but the thermal diffusion occurs along the direction of gas flow, and thus along the temperature gradient. 

Tin will protect copper from corrosion by neutral water. Pure tin is anodic to copper, and protects discontinuities by sacrificial corrosion. Both intermetallic phases are strongly cathodic to copper, and corrosion is stimulated at gaps in wholly alloyed coatings. An adequate thickness of tin is needed for long service, e.g. 25-50 xm. Another diffusion problem occurs with tin-plated brass. Zinc passes very quickly to the tin surface, where under conditions of damp storage zinc corrosion products produce a film... 

An interesting approach to the inter-diffusion problem was made by Rhys S who protected ruthenium-rich ruthenium-gold alloys by palladium-... 

Clay-catalyzed asymmetric Diels-Alder reactions were investigated by using chiral acrylates [10]. Zn(II)- and Ti(IV)-K-10 montmorillonite, calcined at 55 °C, did not efficiently catalyze the cycloadditions of cyclopentadiene (1) with acrylates that incorporate large-size chiral auxiliaries such as cA-3-neopentoxyisobornyl acrylate (2) and (-)-menthyl acrylate (3, R = H) (Figure 4.1). This result was probably due to diffusion problems. 

Copper is intrinsically a better metal than aluminum for the metallization of IC s. Latest developments in MOCVD show that it can be readily deposited without major changes in existing processing equipment. Diffusion problems are minimized and it appears that present barrier materials, such as titanium nitride or titanium-tungsten alloys, should provide adequate diffusion barriers for the copper-silicon couple, certainly up to the highest temperatures presently used in IC s processing (see Ch. 6). The development of CVD copper for semiconductor metallization is on a considerable scale at this time.Clt ]... 

Vabishchevich, P. (1994) Monotone difference schemes for convection-diffusion problems. Differential Equations, 30, 503-531 (in Russian). 

The results obtained in reactions involving the two first examples showed a reduced catalytic activity compared to the homogeneous catalyst, a situation that may be due to diffusion problems. Enantioselectivity was similar or slightly lower than in solution, with 80% ee [21] and 58% ee [22] in the epox-idation of ds-/l-methylstyrene with NaOCl providing the best results. Only in the last example was an improvement in enantioselectivity reported from 51% to 91% ee in the epoxidation of a-methylstyrene. Recovery of the catalyst was only considered in one case [21] and a significant decrease in enantioselectivity was observed on reuse. 

Neal and Nader [260] considered diffusion in homogeneous isotropic medium composed of randomly placed impermeable spherical particles. They solved steady-state diffusion problems in a unit cell consisting of a spherical particle placed in a concentric shell and the exterior of the unit cell modeled as a homogeneous media characterized by one parameter, the porosity. By equating the fluxes in the unit cell and at the exterior and applying the definition of porosity, they obtained... 

Dining the last couple of years CdS-containing Nafion membranes have been apphed for the photocleavage of H2S . They are not comparable with the monograin membranes because the CdS particles are at randomly distributed in a rather thick Nafion membrane. This technique is attractive for some applications because the semiconductor particles are immobilized . On the other hand, problems may arise because of diffusion problems in the nafion membrane. Mainly the photoassistol Hj-formation at CdS was investigated in the presence of a Pt-catalyst and with coprecipitated ZnS CdS without a catalyst . 

The mathematical formulations of the diffusion problems for a micropippette and metal microdisk electrodes are quite similar when the CT process is governed by essentially spherical diffusion in the outer solution. The voltammograms in this case follow the well-known equation of the reversible steady-state wave [Eq. (2)]. However, the peakshaped, non-steady-state voltammograms are obtained when the overall CT rate is controlled by linear diffusion inside the pipette (Fig. 4) [3]. 

The diffusion problem can be solved to predict that one wave in NPV or one pair of anodic and cathodic peak currents in CV are to be observed the half-wave potential is expressed by... 

However, diffusion of the reactive QM out of the enzyme active site is a major concern. For instance, a 2-acyloxy-5-nitrobenzylchloride does not modify any nucleophilic residue located within the enzyme active site but becomes attached to a tryptophan residue proximal to the active site of chymotrypsin or papain.23,24 The lack of inactivation could also be due to other factors the unmasked QM being poorly electrophilic, active site residues not being nucleophilic enough, or the covalent adduct being unstable. Cyclized acyloxybenzyl molecules of type a could well overcome the diffusion problem. They will retain both the electrophilic hydroxybenzyl species b, and then the tethered QM, in the active site throughout the lifetime of the acyl-enzyme (Scheme 11.1). This reasoning led us to synthesize functionalized... 

Two examples will now be given of solution of the convective diffusion problem, transport to a rotating disk as a stationary case and transport to a growing sphere as a transient case. Finally, an engineering approach will be mentioned in which the solution is expressed as a function of dimensionless quantities characterizing the properties of the system. 

Nonequilibrium thermodynamics provides a second approach to combined convection and diffusion problems. The Kedem-Katchalsky equations, originally developed to describe combined convection and diffusion in membranes, form the basis of this approach [6,7] ... 

An overview of this kind is, of necessity, limited in detail. Readers interested in a more thorough development of mass transfer principles are encouraged to consult the references listed at the end of the chapter. In particular, Cussler s excellent textbook on diffusion is an accessible introduction to the subject geared toward the physical scientist [11], Those with a more biological orientation may prefer Friedman s text on biological mass transfer [12], which is also exceptional. A classic reference in the field is Crank s Mathematics of Diffusion [13], which contains solutions to many important diffusion problems. 

Fick s first law is a concise mathematical statement however, it is not directly applicable to solutions of most pharmaceutical problems. Fick s second law presents a more general and useful equation in resolving most diffusion problems. Fick s second law can be derived from Fick s first law. 

In many diffusion problems of practical importance in the pharmaceutical sciences, such as intrinsic dissolution studies and drug release from solid dosage forms, the medium under consideration is not at rest. In addition to concentration changes due to diffusion, there are concentration changes by convection. External forces, such as pressure gradients and temperature differences, can cause convective flows. Although convection can also be caused by diffusion itself, our discussion is limited to convection caused by external forces, since convection produced by diffusion is negligible (less than 10%) for most pharmaceutical problems. 

Membrane diffusion represents over 80% of the diffusion problems of pharmaceutical interest. Membrane diffusion may be called diffusion in a plane sheet, since it involves one-dimensional diffusion in a medium bounded by two parallel... 

Membrane diffusion illustrates the uses of Fick s first and second laws. We discussed steady diffusion across a film, a membrane with and without aqueous diffusion layers, and the skin. We also discussed the unsteady diffusion across a membrane with and without reaction. The solutions to these diffusion problems should be useful in practical situations encountered in pharmaceutical sciences, such as the development of membrane-based controlled-release dosage forms, selection of packaging materials, and experimental evaluation of absorption potential of new compounds. Diffusion in a cylinder and dissolution of a sphere show the solutions of the differential equations describing diffusion in cylindrical and spherical systems. Convection was discussed in the section on intrinsic dissolution. Thus, this chapter covered fundamental mass transfer equations and their applications in practical situations. 

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