Dispersion in a capillary

On the basis of a number of assumptions, Mazo (1998) derives an expression for the effective dispersion coefficient in terms of the velocity profile, system geometry, etc., that reduces to Taylor s (1953) formulation for dispersion in a capillary (i.e., where the dimensionless velocity distribution is given by Eq. [15]). Using this approach dispersion for other velocity profiles can be calculated, although no other examples are presented. 

Gupta, V.K., and R.N. Bhattacharya. 1983. A new derivation of the Taylor-Aris theory of solute dispersion in a capillary. Water Resour. Res. 19 945-951. 

At large Peclet numbers where diffusive mixing is dominant, as for dispersion in a capillary tube, from Eq. (4.6.35)... 

At the present time there exist no flux relations wich a completely sound cheoretical basis, capable of describing transport in porous media over the whole range of pressures or pore sizes. All involve empiricism to a greater or less degree, or are based on a physically unrealistic representation of the structure of the porous medium. Existing models fall into two main classes in the first the medium is modeled as a network of interconnected capillaries, while in the second it is represented by an assembly of stationary obstacles dispersed in the gas on a molecular scale. The first type of model is closely related to the physical structure of the medium, but its development is hampered by the lack of a solution to the problem of transport in a capillary whose diameter is comparable to mean free path lengths in the gas mixture. The second type of model is more tenuously related to the real medium but more tractable theoretically. 

Giddings [2] estimated that, for a well-packed column, (y) takes a value of about 0.6. Equation (11) accurately describes longitudinal dispersion in GC capillary columns and equation (12) accurately describes longitudinal dispersion in GC and LC packed columns. Experimental support for these equations will be given in a later chapter. 

The influence of the vi.scosity ratio 8 on the flow behavior in a capillary was discussed by Rumscheidt and Mason [lOj. They pointed out that when the viscosity ratio is small, the dispersed droplets are drawn out to great lengths but do not burst, and when the viscosity ratio is of the order of unity, the extended droplets break up into smaller droplets. At very high viscosity ratios, the droplets undergo only very limited deformations. This mechanism can explain our observations and supports our theoretical analysis assumptions, summarized previously as points 2, 3, and 4. 

The form of the effective mobility tensor remains unchanged as in Eq. (125), which imphes that the fluid flow does not affect the mobility terms. This is reasonable for an uncharged medium, where there is no interaction between the electric field and the convective flow field. However, the hydrodynamic term, Eq. (128), is affected by the electric field, since electroconvective flux at the boundary between the two phases causes solute to transport from one phase to the other, which can change the mean effective velocity through the system. One can also note that even if no electric field is applied, the mean velocity is affected by the diffusive transport into the stationary phase. Paine et al. [285] developed expressions to show that reversible adsorption and heterogeneous reaction affected the effective dispersion terms for flow in a capillary tube the present problem shows how partitioning, driven both by electrophoresis and diffusion, into the second phase will affect the overall dispersion and mean velocity terms. 

Flow properties of macroemulsions are different from those of non-emulsified phases 19,44). When water droplets are dispersed in a non-wetting oil phase, the relative permeability of the formation to the non-wetting phase decreases. Viscous energy must be expended to deform the emulsified water droplets so that they will pass through pore throats. If viscous forces are insufficient to overcome the capillary forces which hold the water droplet within the pore body, flow channels will become blocked with persistent, non-draining water droplets. As a result, the flow of oil to the wellbore will also be blocked. 

Figure 3. Consistency curve of 13 vol. % graphite dispersed in a water gel using a capillary viscometer...

Figure 1.180 (a) Mixing in a capillary tube by Taylor dispersion. 

A particle or droplet-sizing technique in which the flow of dispersed species in a capillary, between charged electrodes, causes changes in conductivity that are interpreted in terms of the sizes of the species. Coulter is the brand name for the automated counter. See also Sensing Zone Technique. 

The situation shown in Figure lc shows mass transfer from wide to narrow ones cages, which can be found at the evaporation interface. This effect explains the redistribution of the deposited component. The less adhesion the salt has to the surface of a non-uniform capillary, the higher the heterogeneity of its distribution in a capillary. To intensify or decrease mass transfer, and thus to control the dispersion and distribution of deposited compound, one can reasonably manipulate experimental conditions such as viscosity, surface tension, rate of evaporation, and addition of compounds that compete with the main adsorber during adsorption on the surface, etc.4... 

Foam (5) is a collection of gas bubbles with sizes ranging from microscopic to infinite for a continuous gas path. These bubbles are dispersed in a connected liquid phase and separated either by lamellae, thin liquid films, or by liquid slugs. The average bubble density, related to foam texture, most strongly influences gas mobility. Bubbles can be created or divided in pore necks by capillary snap-off, and they can also divide upon entering pore branchings (5). Moreover, the bubbles can coalesce due to instability of lamellae or change size because of diffusion, evaporation, or condensation (5,8). Often, only a fraction of foam flows as some gas flow is blocked by stationary lamellae (4). 

Curl and co-workers (R15, R16, V2) forced the stabilized dispersion through a capillary, where a specially designed photometer assembly (shown in Fig. 4) measured both the drop size distribution and the dye concentration of the drops. Sedimentation techniques have also been used for measuring the drop size distribution (BI3, Gil, Kl, S4). 

In the PCA process originally described by Dixon et al. (19), a small volume of a liquid solution is dispersed through a capillary in a chamber filled with a compressed gas, which can be either subcritical or supercritical. In the original paper, PCA appears as a batch process nevertheless in most articles thereafter, PCA was mainly used when the continuum was subcritical. Subsequently, the use of a coaxial nozzle was suggested to improve the control of particle morphology (20). 

Hoagland and Prud homme (1985) presented a method of moments analysis of dispersion in a single sinusoidal capillary tube with solid walls (Fig. 3-6C). Their geometric model is defined by the wave length (k), the amplitude (a) and mean radius () of the sine wave (z) that describes the aperture wall ... 

Mazo (1998) points out that when the time is sufficiently long that diffusing particles can sample the entire velocity distribution (e.g., the entire cross-section perpendicular to flow in a capillary tube), the velocity distribution of each particle is the same as that of the distribution of velocities in the flow. In an unbounded flow regime, an infinite time would be required and anomalous dispersion results. 

Let us now discuss in some detail the peculiarities of particle motion during electrophoresis and some other electrical properties of free disperse systems. Electrophoresis usually takes place in a stationary liquid. In a moving fluid the motion of particles occurs only in thin flat gaps and capillaries (microelectrophoresis), where the fluid motion is caused by electroosmosis. If fairly large non-conducting particles are dispersed in a rather dilute electrolyte solution, the ratio of particle radius to the double layer thickness may be substantially greater than 1, i.e., r/8 = kt 1. The streamlines of outer electric field surround the particle and are parallel to most of its surface, as shown in Fig. V-9. In this case the particle velocity, v0, can be with good precision described by Helmholtz-Smoluchowski equation. 

Table 7.6 provides a partial reference to studies on the effects of flow on the morphology of polymer blends [Lohfink, 1990 Walling, 1995]. Dispersed phase morphology development has been mainly studied in a capillary flow. To explain the fibrillation processes, not only the viscosity ratio, but also the elasticity effects and the interfacial properties had to be considered. In agreement with the microrheology of Newtonian systems, an upper bound for the viscosity ratio, X, has also been reported for polymer blends — above certain value of X (which could be significantly larger than the... 

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