Diffusion dispersion equation

The root time method of data analysis for diffusion coefficient determination was developed by Mohamed and Yong [142] and Mohamed et al. [153]. The procedure used for computing the diffusion coefficient utilizes the analytical solution of the differential equation of solute transport in soil-solids (i.e., the diffusion-dispersion equation) ... 

The chromatography literature contains a vast amount of dispersion data for all types of chromatography and, in particular, much of the data pertains directly to GC and LC. Unfortunately, almost all the data is unsuitable for validating one particular dispersion equation as opposed to another. There are a number of reasons for this firstly, the necessary supporting data (e.g., diffusivity data for the solutes in the solvents employed as the mobile phase, accurate distribution and/or capacity factor constants (k")) are not available secondly, the accuracy and precision of much of the data are inadequate, largely due to the use of inappropriate apparatus with high extracolumn dispersion. 

C. In their first series of experiments, six data sets were obtained for (H) and (u), employing six solvent mixtures, each exhibiting different diffusivities for the two solutes. This served two purposes as not only were there six different data sets with which the dispersion equations could be tested, but the coefficients in those equations supported by the data sets could be subsequently correlated with solute diffusivity. The solvents employed were approximately 5%v/v ethyl acetate in n-pentane, n-hexane, n-heptane, -octane, -nonane and n-decane. The solutes used were benzyl acetate and hexamethylbenzene. The diffusivity of each solute in each solvent mixture was determined in the manner of Katz et al. [3] and the values obtained are included... 

Dispersion equations, typically the van Deemter equation (2), have been often applied to the TLC plate. Qualitatively, this use of dispersion equations derived for GC and LC can be useful, but any quantitative relationship between such equations and the actual thin layer plate are likely to be fraught with en or. In general, there will be the three similar dispersion terms representing the main sources of spot dispersion, namely, multipath dispersion, longitudinal diffusion and dispersion due to resistance to mass transfer between the two phases. 

HETP of a TLC plate is taken as the ratio of the distance traveled by the spot to the plate efficiency. The same three processes cause spot dispersion in TLC as do cause band dispersion in GC and LC. Namely, they are multipath dispersion, longitudinal diffusion and resistance to mass transfer between the two phases. Due to the aforementioned solvent frontal analysis, however, neither the capacity ratio, the solute diffusivity or the solvent velocity are constant throughout the elution of the solute along the plate and thus the conventional dispersion equations used in GC and LC have no pertinence to the thin layer plate. 

In considering axial dispersion as a diffusive flow, we assume that Fick s first law applies, with the diffusion or effective diffusion coefficient (equation 8.5-4) replaced by an axial dispersion coefficient, D,. Thus, for unsteady-state behavior with respect to a species A (e.g., a tracer), the molar flux (NA) of A at an axial position x is... 

The EOT flow has a flat profile, almost as a piston therefore, its contribution to dispersion of the migrating zones is small. The EOT positively contributes to the axial diffusion variance (Equation (31)), when it moves in the same direction as the analytes. However, when the capillary wall is not uniformly charged, local turbulence may occur and cause irreproducible dispersion. ... 

While the advection-dispersion equation has been used widely over the last half century, there is now widespread recognition that this equation has serious limitations. As noted previously, laboratory and field-scale application of the advection-dispersion equation is based on the assumption that dispersion behaves macroscopically as a Fickian diffusive process, with the dispersivity being assumed constant in space and time. However, it has been observed consistently through field, laboratory, and Monte Carlo analyses that the dispersivity is not constant but, rather, dependent on the time or length scale of measurement (Gelhar et al. 1992),... 

To quantify such transport, the advection-dispersion equation, which requires a narrow pore-size distribution, often is used in a modified framework. Van Genuchten and Wierenga (1976) discuss a conceptualization of preferential solute transport throngh mobile and immobile regions. In this framework, contaminants advance mostly through macropores containing mobile water and diffuse into and out of relatively immobile water resident in micropores. The mobile-immobile model involves two coupled equations (in one-dimensional form) ... 

Regardless of the transport equation considered, the major effect of sorption on contaminant breakthrough curves is to delay the entire curve on the time axis, relative to a passive (nonsorbing) contaminant. Because of the longer residence time in the porous medium, advective-diffusive-dispersive interactions also are affected, so that longer (non-Fickian) tailing in the breakthrough curves is often observed. 

In this situation, transport equations similar to those discussed previously can be applied. For example, by assuming sorption to be essentially instantaneous, the advective-dispersion equation with a reaction term (Saiers and Hornberger 1996) can be considered. Alternatively, CTRW transport equations with a single ti/Ci, t) can be applied or two different time spectra (for the dispersive transport and for the distribution of transfer times between mobile and immobile—diffusion, sorption— states can be treated Berkowitz et al. 2008). 

As the alternative, a phenomenological description of turbulent mixing gives good results for many situations. An apparent diffusivity is defined so that a diffusion-type equation may be used, and the magnitude of this parameter is then found from experiment. The dispersion models lend themselves to relatively simple mathematical formulations, analogous to the classical methods for heat conduction and diffusion. 

Let US return to the discussion of computational transport routines, where each computational cell is the equivalent of a complete mix reactor. If we are putting together a computational mass transport routine, we could simply specify the size of the cells to match the diffusion/dispersion in the system. The number of well-mixed cells in an estuary or river, for example, could be calculated from equation (6.44), assuming a small Courant number. Then, the equivalent longitudinal dispersion coefficient for the system would be calculated from equation (6.44), as well, for a small At (At was infinitely small in equation 6.44) ... 

A somewhat academic problem also seems to have found its solution in the work of Nauman (57) who showed that RTD in systems governed by the dispersion equation could be defined as in closed systems provided the time spent by the tracer outside the reactor owing to diffusion be excluded from the count. 

In this equation, represents the effective lateral diffusion/dispersion constant for laminar flow a value on the order of is suitable, and for turbulent flow the molecular diffusion coefficient in this expression should be replaced by the turbulent diffusion coefficient. Based on these simple relations it can be calculated that under typical conditions, lateral reactant transport takes place only over distances of a few subchannels. In other words, if the gas velocity through the wall channels differs much from the velocity through the central subchannels, the nonuniform flow profile can have a significant effect on the overall reactant conversion. In these situations the CBS model can be expected to give a better estimate of the reactor performance than the CB model. 

The first 3D model of FFF was developed in Ref. 2. The 3D diffusion-convection equation was solved with the help of generalized dispersion theory, resulting in the equations for the cross-sectional average concentration of the solute and dispersion coefficients and K2, representing the normalized solute zone velocity and the velocity of the corresponding peak width growth, respectively. Unfortunately, only the steady-state asymptotic values of dispersion coefficients Ki oo) and K2 oo) were determined in Ref. 2, leading to the prediction of the solute peaks much wider than the experimental ones. 

A more general case was studied in Refs. 4 and 5, where different initial solute distributions were examined, including the distributions describing the syringe inlet with and without the stop-flow relaxation. The 3D generalized dispersion theory was used to solve the 3D diffusion-convection equation. Unlike Refs. 2 and 3, the exact equation for flow profile in the rectangular channel was used ... 

So far, the concept of mass conservation has been applied to large, easily measurable control volumes such as lakes. Mass conservation also can be usefully expressed in an infinitesimal control volume, mathematically considered to be a point. Conservation of mass is expressed in such a volume with the advection—dispersion-reaction equation. This equation states that the rate of change of chemical storage at any point in space, dC/dt, equals the sum of both the rates of chemical input and output by physical means and the rate of net internal production (sources minus sinks). The inputs and outputs that occur by physical means (advection and Fickian transport) are expressed in terms of the fluid velocity (V), the diffusion/dispersion coefficient (D), and the chemical concentration gradient in the fluid (dC/dx). The input or output associated with internal sources or sinks of the chemical is represented by r. In one dimension, the equation for a fixed point is... 

If transport occurs much faster than sorption, sorption processes may not reach equilibrium conditions. Nonequilibrium sorption may result from physical causes such as intraparticle rate-limited diffusion, chemical causes such as rate-limiting reaction kinetics, or a combination of the two. One approach used to model rate-limited sorption is bi-continuum models consisting of one region where transport is described by the advection-dispersion equation with equilibrium sorption, and another region where transport is diffusion limited with equilibrium sorption, or another region where sorption is chemically rate limited. 

Equation 2.20 is the advection-dispersion (AD) equation. In the petroleum literature, the term convection-diffusion (CD) equation is used, or simply diffusion equation (Brigham, 1974). When a reaction term is included, the term advection-reaction-dispersion (ARD) equation is used elsewhere. When the adsorption term is expressed as a reaction term, the ARD equation is as discussed later in Section 2.4. Several solutions of Eq. 2.20 have been presented in the literature, depending on the boundary conditions imposed. In general, they are various combinations of the error function. When the porous medium is long compared with the length of the mixed zone, they all give virtually identical results. 

Several parameters have been used to gather information about sample dispersion in flow analysis peak variance [109], time of appearance of the analytical signal, also known as baseline-to-baseline time [110], number of tanks in the tanks-in-series model [111], the Peclet number in the axially dispersed plug flow model [112], the Peclet number and the mean residence time in the diffusive—convective equation [113]. 

As the sample plug starts being pushed forward, axial diffusion is the main component of the dispersion process, due to the high concentration gradients at the sample/carrier stream interface. The hypothetical peak shape associated with the flowing sample is shown in Fig. 5.9b, which corresponds to the first theoretical Taylor solution [28,29] for the diffusive-convective equation (Eq. 3.4). Situations associated with Fig. 5.9a,b never occur in practice in flow injection analysis. 

We stress that all the approaches presented in this subsection, leading to normal and anomalous diffusion, are intended to model dispersion. As clearly seen in (2.28) and their equivalents for anomalous diffusion these equations damp the higher Fourier modes so that there is a relative increase of variance at larger scales. But in situations such as in modelling dispersion induced by turbulent flows the dispersion is a consequence of the stretching of fluid elements that is associated with compression along the complementary directions. This produces a direct cascade of variance towards small scales that is not captured by the diffusion equation (2.9). Thus the small scale distribution and phenomena strongly dependent on this, like chemical reactivity, may not be adequately modelled in the framework of diffusion equations alone. 

Equation (2.87) represents the competition between the convergent component of the advecting flow, that compresses the concentration towards the origin, and diffusive dispersion. Since this simple model equation aims only to describe the dominant processes, we neglect secondary effects like the fluctuations of the stretching rate and curvature effects due to the folding of the filament. We... 

Movement and fate of radionuclides in groundwater follow the transport components represented by the basic diffusion/dispersion-advection equation. The following expression describes the basic equation for the advective and dispersive transport with radioactive decay and retardation for the radionuclide transport in the groundwater ... 

For a membrane column, it is essential that axial backmixing by diffusion (dispersion) be negligible compared to convective transport. In this case, Equation 34 is reduced to ... 

Infiltration with diffusion/dispersion. In real flow systems, both diffusion and dispersion accompany fluid flow. The general equation of open system fluid-rock interaction is Equation (20). It has an infiltration term, together with a dispersion/diffusion term. Fluid-rock exchange is generally described by a kinetic formulation. Equation (30) is used here, for illustrative purposes. Substituting Equation (30) into Equation (20) one obtains ... 

The general kinetic-dispersion-diffusion-infiltration equation. Equations (47) and (48) can be inserted into Equation (46), along with the definitions of dimensionless distance and time ... 

More complicated reactions can be easily treated by the methods outlined in the preceeding sections, that is (a) determine the coupled diffusion-chemical reaction equations, (b) linearize the equations in the concentration fluctuations, (c) solve the linearized rate equations by Fourier-Laplace transforms, (d) solve the dispersion equation... 

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