Diffusion-convection process compared

Permeation of mAbs across the cells or tissues is accomplished by transcellular or paracellular transport, involving the processes of diffusion, convection, and cellular uptake. Due to their physico-chemical properties, the extent of passive diffusion of classical mAbs across cell membranes in transcellular transport is minimal. Convection as the transport of molecules within a fluid movement is the major means of paracellular passage. The driving forces of the moving fluid containing mAbs from (1) the blood to the interstitial space of tissue or (2) the interstitial space to the blood via the lymphatic system, are gradients in hydrostatic pressure and/or osmotic pressure. In addition, the size and nature of the paracellular pores determine the rate and extent of paracellular transport. The pores of the lymphatic system are larger than those in the vascular endothelium. Convection is also affected by tortuosity, which is a measure of hindrance posed to the diffusion process, and defined as the additional distance a molecule must travel in a particular human fluid (i. e., in vivo) compared to an aqueous solution (i. e., in vitro). 

This paper reports the mathematical modelling of electrochemical processes in the Soderberg aluminium electrolysis cell. We consider anode shape changes, variations of the potential distribution and formation of a gaseous layer under the anode surface. Evolution of the reactant concentrations is described by the system of diffusion-convection equations while the elliptic equation is solved for the Galvani potential. We compare its distribution with the C02 density and discuss the advantages of the finite volume method and the marker-and-cell approach for mathematical modelling of electrochemical reactions. 

Eq. (34) also shows that inclined surfaces should yield slowly, decreasing diffusive transport capabilities compared to the spinning disk, owing to the thicker films. However, for reactions with some kinetic limitations the increased residence time may be more beneficial. It should also be noted that the results of mixing by collisions, motion over structured surfaces and through surface waves, are convective processes not considered in this formulation. 

Sion increases an this step, we have to explain how can the concentration of C,C1 and Cj Pi vary during the transfer. Several authors [ 21,22,Ib,23] studying similar interface have shown that the adsorption -desorption process is very fast as compared to diffusion convection. This implies that, at every time, the adsorbed monolayer is in equilibrium with the subjacent bulk layers and diffusion and convection are the limiting steps to the present transfer. 

It is a typical feature of the diffusion processes at electrodes of small size, which are reached by converging diffusion fluxes, that a steady state can be attained even without convection (e.g., in gelled solutions). Such electrodes, which have dimensions comparable to typical values of 8, are called microelectrodes. 

Most of the models assume that neutral-species transport can be represented with either a well-mixed model or a plug flow model. The major drawback to these assumptions is that important inelastic rate processes such as molecular dissociation are usually localized in space in the reactor and are often fast compared with rates of diffusion or convection. As a result, the spatial variation of fluid flow in the reactor must be accounted for. This variation introduces a major complication in the model, because the solution of the nonisothermal Navier-Stokes equations in multidimensional geometries is expensive and difficult. 

Two limiting situations may be identified r (1) the rate constant K is very small compared to aD, hence the process occurring in the interaction forces boundary layer controls the deposition rate, and (2) the rate constant K is very large hence the convective diffusion is the controlling factor. The first limiting case was treated by Hull and Kitchener (except for the variation of the diffusion coefficient) while the second was treated by Levich. In the present paper an equation is established which is valid for all values of the rate constant thus also incorporating both limiting situations. 

Boundary layer formulation. Many membrane processes are operated in cross-flow mode, in which the pressurised process feed is circulated at high velocity parallel to the surface of the membrane, thus limiting the accumulation of solutes (or particles) on the membrane surface to a layer which is thin compared to the height of the filtration module [2]. The decline in permeate flux due to the hydraulic resistance of this concentrated layer can thus be limited. A boundary layer formulation of the convective diffusion equation can give predictions for concentration polarisation in cross-flow filtration and, therefore, predict the flux for different operating conditions. Interparticle force calculations are used in two ways in this approach. Firstly, they allow the direct calculation of the osmotic pressure at the membrane. This removes the need for difficult and extensive experi-... 

Here, C( , z, t) is the scaled solute concentration in the fluid phase, Cw the solute concentration at the wall, 6 the normalized adsorbed concentration (O<0< 1), K the adsorption equilibrium constant, p the transverse Peclet number, T represents the adsorption capacity (ratio of adsorption sites per unit tube volume to the reference solute concentration), and Da is the local Damkohler number (ratio of transverse diffusion time to the characteristic adsorption time). We shall assume that p 4Cl while T and Da are order-one parameters. (In physical terms, this implies that transverse molecular diffusion and adsorption processes are much faster compared to the convection.)... 

Since equations (1), (2a), and (3) are formally the same, it is necessary to find criteria to distinguish the three cases. This is possible because 6 varies so much with stirring speed, and the diffusion coefficient D is affected by viscosity while the parameters in chemical rates usually are not. Different metals dissolve at the same rate in the same solution if the rate is controlled by convection-diffusion. The activation energy of diffusive transport, as measured from temperature coefficients, is normally much lower than the activation energy of chemical processes (3000-6000 cal/mole compared to 10,000-20,000 cal/mole, although some chemical reactions do have lower values). 

Compared to this idealized model, the actual flux of Rn may be diminished by the saturation of pore space by water (the mean length of Rn diffusion in water is on the order of a milhmeter, so saturation diminishes the flux by up to a factor of 1,000) and decreases in porosity with depth. Advection of gas through soil in response to barometric pressure change, soil gas convection, and transpiration of Rn saturated soil solution will increase the radon escape rate. All of these processes are difficult to model accurately, so the determination of Rn fluxes rehes on measurements. 

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