Convection diffusion

Reviews of concentration polarization have been reported (14,38,39). Because solute wall concentration may not be experimentally measurable, models relating solute and solvent fluxes to hydrodynamic parameters are needed for system design. The Navier-Stokes diffusion—convection equation has been numerically solved to calculate wall concentration, and thus the water flux and permeate quaUty (40). 

A simplified model usiag a stagnant boundary layer assumption and the one-dimension diffusion—convection equation has been used to calculate wall concentration ia an RO module. The iategrated form of this equation, the widely appHed film theory (41), is given ia equation 8. 

In the film-penetration model (H19), it is assumed that the reactant A penetrates through the surface element by one-dimensional unsteady-state molecular diffusion. Convective transport is assumed to be insignificant. The diffusing stream of the reactant A is depleted along the path of diffusion by its reversible reaction with the reactant B, which is an existing component of the liquid surface element. If such a reaction can be represented as... 

Fluid flow and reaction engineering problems represent a rich spectrum of examples of multiple and disparate scales. In chemical kinetics such problems involve high values of Thiele modulus (diffusion-reaction problems), Damkohler and Peclet numbers (diffusion-convection-reaction problems). For fluid flow problems a large value of the Mach number, which represents the ratio of flow velocity to the speed of sound, indicates the possibility of shock waves a large value of the Reynolds number causes boundary layers to be formed near solid walls and a large value of the Prandtl number gives rise to thermal boundary layers. Evidently, the inherently disparate scales for fluid flow, heat transfer and chemical reaction are responsible for the presence of thin regions or "fronts in the solution. 

Combining hindered diffusion theory with the diffusion/convection problem in the model pore, Trinh et al. [399] showed how the effective transport coefficients depend upon the ratio of the solute to pore size. Figure 28 shows that as the ratio of solute to pore size approaches unity, the effective mobility function becomes very steep, thus indicating that the resolution in the separation will be enhanced for molecules with size close to the size of the pore. Similar results were found for the effective dispersion, and the implications for the separation of various sizes of molecules were discussed by Trinh et al. [399]. 

Trinh, S Locke, BR Arce, P, Diffusive-Convective and Diffusive-Elechoconvective Transport in Non-Uniform Channels with Application to Macromolecular Separations, Separation and Purification Technology 15, 255, 1999. 

The movement of gases through the atmosphere, such as the fragrance of a rose moving from the flower to our noses, generally involves air currents as well as diffusion. Convection, the flow of gas in a current, moves molecules much more rapidly than... 

Aeis, R., On the dispersion of a solute by diffusion, convection and exchange between phases, Proc. R. Soc. London, A 252 (1959) 538-550. 

In practice, the Peclet number can always be ignored in the diffusion-convection equation. It can also be ignored in the root boundary condition unless C > X/Pc or A, < Pe. Inspection of the table of standard parameter values (Table 2) shows that this is never the case for realistic soil and root conditions. Inspection of Table 2 also reveals that the term relating to nutrient efflux, e, can also be ignored because e < Pe [Pg.343]

This chapter provides analytical solutions to mass transfer problems in situations commonly encountered in the pharmaceutical sciences. It deals with diffusion, convection, and generalized mass balance equations that are presented in typical coordinate systems to permit a wide range of problems to be formulated and solved. Typical pharmaceutical problems such as membrane diffusion, drug particle dissolution, and intrinsic dissolution evaluation by rotating disks are used as examples to illustrate the uses of mass transfer equations. 

The reaction plane model with heterogeneous reactions was discussed at length for acid-base reactions in the previous section. The same modeling technique, of confining the reactions to planes, can be applied to micelle-facilitated dissolution. As with the acid-base model, one starts with a one-dimensional steady-state equation for mass transfer that includes diffusion, convection, and reaction. This equation is then applied to the individual species i, i.e., the solute, s, the micelle, m, and the drug-loaded micelle, sm, to yield... 

There are, in principle, three ways in which material may be transported to the electrode surface diffusion, convection and migration. Of these, perhaps the most straightforward is migration, which simply consists of the movement of a charged particle under the influence of an electric field. Experimentally, it is well established that after an extremely short time an ion in solution in an electric field will behave as if it had acquired a steady velocity in the direction of the field. The reason why a steady velocity is established rather... 

The introduction of Equation (9.71) for Equation (9.26e) makes this a new problem identical to what was done for the pure diffusion/convective modeling of the burning rate. Hence L is simply replaced by L,n to obtain the solution with radiative effects. Some rearranging of the stagnant layer case can be very illustrative. From Equations (9.61) and (9.42) we can write... 

The movement of synthetic pyrethroids in soil and sediment is basically controlled by diffusion, convection, and dispersion. When entering a water-sediment system... 

To know that the overall process of mass transport occurs via three mechanisms, namely convection, migration and diffusion. Convection is the physical movement of solution, migration is the movement of charged analyte in response to Coulomb s law and diffusion is an entropy-driven process. In terms of mass transport, the order of effectiveness is as follows convection migration > diffusion. 

Mass transport comprises three different modes , i.e. convection, migration and diffusion. Convection (stirring) is the most efficient form of mass transport. Migration can be minimized by adding a swamping electrolyte to the solution. Diffusion occurs even in the absence of migration and convection. 

What are the mechanisms and the transport coefficients of water fluxes (diffusion, convection, hydraulic permeation, electro-osmotic drag) ... 

The advection—diffusion equation with a source term can be solved by CFD algorithms in general. Patankar provided an excellent introduction to numerical fluid flow and heat transfer. Oran and Boris discussed numerical solutions of diffusion—convection problems with chemical reactions. Since fuel cells feature an aspect ratio of the order of 100, 0(100), the upwind scheme for the flow-field solution is applicable and proves to be very effective. Unstructured meshes are commonly employed in commercial CFD codes. 

Liang Y. (1994) Axisymmetric double-diffusive convection in a cylindrical container linear stability analysis with applications to molten Ga0-Al203-Si02. In Double-... 

The aim of the article is to introduce new observations of diffusive-convective phenomena in polymer chemistry. The processes discussed are of significance to those interested in transport phenomena. 

Nucleation of the chalcogenide is much simpler in this process, since a solid phase—the metal hydroxide (or other solid phase)—is already present and the process proceeds by a substitution reaction on that solid phase. In this case, the initial step in the deposition is adhesion of the hydroxide to the substrate. This hydroxide is then converted into, e.g., CdS, forming a primary deposit of CdS clusters. More Cd(OFI)2 and, as the reaction proceeds, CdS and partially converted hydroxide diffuses/convects to the substrate, where it may stick, either to uncovered substrate (in the early stages of deposition) or to already deposited material. This is essentially the same process as aggregation, described in Chap-... 

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