Solutions to Transport with Convection

Diffusive transport with convection occurring simultaneously can be solved more easily if we orient our coordinate system properly. First, we must orient one axis in the direction of the flow. In this case, we will choose the v-coordinate so that u is nonzero and v and w are zero. Second, we must assume a uniform velocity profile, u = U = constant with y and z. Then, equation (2.33) becomes 

Determination of Dispersion Coefficient from Tracer Clouds 

The one-dimensional mass transport equation for plug flow with dispersion, and a 

We will convert our fixed coordinate system to a coordinate system moving at velocity U through the change of variables, x = x - Ut. Then, equation (6.39) is given as 

if we are determining the dispersion coefficient through the use of a pulse tracer cloud, the boundary conditions are those of a Dirac delta  

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A second commonly used pore-restriction model is defined by the permeability of a solute ion through a membrane relative to water, using the reflection coefficient, a. It was pointed out by Davson [102] that the reflection coefficient, with limits o = 1, no entry, and a = 0, no restriction on entry, correlates well with the Renkin model. In the present context, 1 - o is simply iont/ watep Where / water the iontophoretic permeability coefficient of water [68]. Plots of log (1 - a) versus r, log MV, or MV should give slopes identical to plots based on The reflective coefficient, a, is often now used to correct for differences in the extent of solute ion transport with convective flow during iontophoresis [68,103,104]. 

Figure E7.3.1. Computational solutions to the purely convective filter transport problem with varying Courant number.

The Staverman reflection coefficient, o, measures the extent to which the membrane rejects a given solute purely transported by convection. Solutes fully rejected by the membrane feature o = 1. Solutes freely permeating the membrane feature ct = 0. Membrane rejection toward a given solute is experimentally assessed in the course of pure filtration experiments in terms of its rejection coefficient R, or its sieving coefficient S, with S = —R being the permeate-to-retentate solute concentration ratio. In fact, R is related to a as follows (Spiegler and Kedem, 1996) ... 

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]. 

Tubes are much more sensitive to convection effects than capillaries, but capillaries contain much smaller amounts of solution for analysis. Transport by convection when tubes are used can be accounted for by experimental evaluation. A disadvantage of both methods is the amount of time required for the experiment, which may be hundreds of hours. The investigator must address the possibility of adsorption of the diffusing solute onto the glass surfaces. In addition, the dimensions of the glass capillary must be known with considerable accuracy. 

When working with a computational transport code, there is httle reason to simplify equation (2.14) further. Our primary task, however, is to develop approximate analytical solutions to environmental transport problems, and we will normally be assuming that diffusion coefficient is not a function of position, or x, y, and z. We can also expand the convective transport terms with the chain rule of partial differentiation ... 

Figure E7.4.1. Solution to convective transport problem at Co = 0.1 and comparison with equation (E7.4.7) with D = U zl2. Cou, Courant number.

What characterises the different incubation steps is the time required to reach thermodynamic equilibrium between an antibody and an antigen in the standard format of microtitre plates. In fact the volume used in each of the incubation steps has been fixed between 100 and 200 pL to be in contact with a surface area of approx. 1 cm2 where the affinity partner is immobilised. The dimensions of the wells are such that the travel of the molecule from the bulk solution to the wall (where the affinity partner is immobilised) is in the order of 1 mm. It must be taken into account that the generation of forced convection or even of turbulence in the wells of a microtitre plate is rather difficult due to the intrinsic dimensions of the wells [10]. Indeed, even if some temperature or shaking effects can help the mass transport from the solution to the wall, the main mass transport phenomenon in these dimensions is ensured by diffusion. 

In this chapter, we present most of the equations that apply to the systems and processes to be dealt with later. Most of these are expressed as equations of concentration dynamics, that is, concentration of one or more solution species as a function of time, as well as other variables, in the form of differential equations. Fundamentally, these are transport (diffusion-, convection-and migration-) equations but may be complicated by chemical processes occurring heterogeneously (i.e. at the electrode surface - electrochemical reaction) or homogeneously (in the solution bulk chemical reaction). The transport components are all included in the general Nernst-Planck equation (see also Bard and Faulkner 2001) for the flux Jj of species j... 

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