Longitudinal transport time

The representative case considered in Taylor (1993) is his case (B), where the longitudinal transport time L/uq is much bigger than the transversal diffusive time fl /D. The problem of a diffusive transport of a solute was studied experimentally and analytically. Two basically different cases were subjected to experimental verification in Taylor s paper ... 

Taking account of the species migration velocity, we slightly modify the above expression to be the time scale ratio between transverse and longitudinal transport ... 

Diffusion plays an important part in peak dispersion. It not only contributes to dispersion directly (i.e., longitudinal diffusion), but also plays a part in the dispersion that results from solute transfer between the two phases. Consider the situation depicted in Figure 4, where a sample of solute is introduced in plane (A), plane (A) having unit cros-sectional area. Solute will diffuse according to Fick s law in both directions ( x) and, at a point (x) from the sample point, according to Ficks law, the mass of solute transported across unit area in unit time (mx) will be given by... 

The area of actual erg and dune formation is delimited by the 150 mm/yr isohyet. This precipitation boundary appears to have shifted strongly in the recent past. Between 20,000 and 13,000 yr BP, the southern limit of active dune formation in the Sahara desert was 800 km south of its present position and most of the now sparely vegetated Sahelian zone was an area of active dune formation at that time. These dunes, mostly of the longitudinal type, are now fixed by vegetation, but their aeolian parentage is still obvious from their well-sorted material. A similar story can be told for the Kalahari sands. Overgrazing in recent times has reactivated aeolian transport in many regions with sands. 

In the ideal plug flow reactor, the flow traverses through the reactor hke a plug, with a uniform velocity profile and no diffusion in the longitudinal direction, as illustrated in Figure 6.2. A nonreactive tracer would travel through the reactor and leave with the same concentration versus time curve, except later. The mass transport equation is... 

In porous media the flow of water and the transport of solutes is complex and three-dimensional on all scales (Fig. 25.1). A one-dimensional description needs an empirical correction that takes account of the three-dimensional structure of the flow. Due to the different length and irregular shape of the individual pore channels, the flow time between two (macroscopically separated) locations varies from one channel to another. As discussed for rivers (Section 24.2), this causes dispersion, the so-called interpore dispersion. In addition, the nonuniform velocity distribution within individual channels is responsible for intrapore dispersion. Finally, molecular diffusion along the direction of the main flow also contributes to the longitudinal dispersion/ diffusion process. For simplicity, transversal diffusion (as discussed for rivers) is not considered here. The discussion is limited to the one-dimensional linear case for which simple calculations without sophisticated computer programs are possible. 

The main components of inner bark are sieve elements, parenchyma cells, and sclerenchymatous cells. Sieve elements perform the function for transportation of liquids and nutrients. More specifically and according to their shape the sieve elements are divided into sieve cells and sieve tubes. The former types are present in gym nosperms, the latter in angiosperms. The sieve elements are arranged in longitudinal cell rows which are connected through sieve areas. The sieve cells are comparatively narrow with tapering ends, whereas the sieve tubes are thicker and cylindrical. After 1 -2 years, or after a longer time in the monocotyledons, the activity of the sieve elements ceases and they are replaced by new elements. 

FIGURE 1-7 Fickian transport by dispersion as water flows through a porous medium such as a soil. Seemingly random variations in the velocity of different parcels of water are caused by the tortuous and variable routes water must follow. This situation contrasts with that of Fig. 1-6, in which turbulence is responsible for the random variability of fluid paths. In this case as well as in the previous one, Fickian mass transport is driven by the concentration gradient and can be described by Fick s first law. The mass transport effect arising from dispersion can be further visualized in Fig. 3-17. There, a mass initially present in a narrow slice in a column of porous media is transported by mechanical dispersion in such a way as to form a wider but less concentrated slice. At the same time, the center of mass also is transported longitudinally in the direction of water flow. 

FIGURE 2-4 Transport of a chemical in a river. At time zero, a pulse injection is made at a location defined as distance zero in the river. As shown in the upper panel, at successive times C, t2, and t3, the chemical has moved farther downstream by advection, and also has spread out lengthwise in the river by mixing processes, which include turbulent diffusion and the dispersion associated with nonuniform velocity across the river cross section. Travel time between two points in the river is defined as the time required for the center of mass of chemical to move from one point to the other. Chemical concentration at any time and distance may be calculated according to Eq. [2-10]. As shown in the lower panel, Cmax, the peak concentration in the river at any time t, is the maximum value of Eq. [2-10] anywhere in the river at that time. The longitudinal dispersion coefficient may be calculated from the standard deviation of the concentration versus distance plot, Eq. [2-7]. 

Simple transport models have significant limitations in a complex estuarine setting commonly, sophisticated numerical models are employed to predict transport in estuaries. In long, narrow estuaries, however, a simple onedimensional model, such as is used in rivers, that incorporates a longitudinal dispersion coefficient and a time-averaged seaward water velocity can be useful. The results of such a model must be averaged over the tidal cycle concentrations at each point in the estuary may be expected to vary significantly with the state of the tide. See Fischer et al. (1979) for a more complete discussion of transport in estuaries. 

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