Dispersion lateral, coefficient

A flow reac tor with some deviation from plug flow, a quasi-PFR, may be modeled as a CSTR battery with a characteristic number n of stages, or as a dispersion model with a characteristic value of the dispersion coefficient or Peclet number. These models are described later. 

Figure 6- The Pasquill-Gifford dispersioo coefficients versus downwind distance for various dispersion classes, (a) The lateral dispersion coefficient and b) the vertical dispersion coefficient are plotted against x.

As was proven later by Bishop [19], the coefficient A in the expansion (73) is the same for all optical processes. If the expansion (73) is extended to fourth-order [4,19] by adding the term the coefficient B is the same for the dc-Kerr effect and for electric field induced second-harmonic generation, but other fourth powers of the frequencies than are in general needed to represent the frequency-dependence of 7 with process-independent dispersion coefficients [19]. Bishop and De Kee [20] proposed recently for the all-diagonal components yaaaa the expansion... 

My interest at that time revolved around evaluating optical rotary dispersion data [12]. The paired values of optical rotation vs. wavelength were used to fit a function called the Drude equation (later modified to the Moffitt equation for William Moffitt [Harvard University] who developed the theory) [13]. The coefficients of the evaluated equation were shown to be related to a significant ultraviolet absorption band of a protein and to the amount of alpha-helix conformation existing in the solution of it. 

Gifford and Hanna tested their simple box model for particulate matter and sulfur dioxide predictions for annual or seasonal averages against diffusion-model predictions. Their conclusions are summarized in Table 5-3. The correlation coefficient of observed concentrations versus calculated concentrations is generally higher for the simple model than for the detailed model. Hanna calculated reactions over a 6-h period on September 30, 1%9, with his chemically reactive adaptation of the simple dispersion model. He obtained correlation coefficients of observed and calculated concentrations as follows nitric oxide, 0.97 nitrogen dioxide, 0.05 and rhc, 0.55. He found a correlation coefficient of 0.48 of observed ozone concentration with an ozone predictor derived from a simple model, but he pointed out that the local inverse wind speed had a correlation of 0.66 with ozone concentration. He derived a critical wind speed formula to define a speed below which ozone prediction will be a problem with the simple model. Further performance of the simple box model compared with more detailed models is discussed later. 

In this model, the effect of the velocity profile is lumped into the dispersion coefficients, as will be discussed later. In comparison, the coefficients calculated from the uniform dispersion model, or the general dispersion model, are more basic in the sense that they do not have two effects combined into one coefficient. 

The data were plotted, as shown in Fig. 11, using the effective diameter of Eq. (50) as the characteristic length. For fully turbulent flow, the liquid and gas data join, although the two types of systems differ at lower Reynolds numbers. Rough estimates of radial dispersion coefficients from a random-walk theory to be discussed later also agree with the experimental data. There is not as much scatter in the data as there was with the axial data. This is probably partly due to the fact that a steady flow of tracer is quite easy to obtain experimentally, and so there were no gross injection difficulties as were present with the inputs used for axial dispersion coefficient measurement. In addition, end-effect errors are much smaller for radial measurements (B14). Thus, more experimentation needs to be done mainly in the range of low flow rates. 

The relations developed for longitudinal dispersion coefficient are given in Table 6.3. The experimental results in rivers tend to have a large range because of the variety of lateral velocity profiles that exist in natural rivers and streams. 

Tracer Determination of Longitudinal Dispersion Coefficient in Rivers. Tracers are generally used to determine longitudinal dispersion coefficient in rivers. Some distance is required, however, before the lateral turbulent diffusion is balanced by longitudinal convection, simitar to Taylor s (1953) analysis of dispersion in a laminar flow. This transport balancing distance, x is given by the equation... 

Once the mathematical description of dispersion has been clarified, we are left with the task of quantifying the dispersion coefficient, Eiis. Obviously, Edh depends on the characteristics of the flow field, particularly on the velocity shear, dvx/dy and dvx /dz. As it turns out, the shear is directly related to the mean flow velocity vx. In addition, the probability that the water parcels change between different streamlines must also influence dispersion. This probability must be related to the turbulent diffusivity perpendicular to the flow, that is, to vertical and lateral diffusion. At this point it is essential to know whether the lateral and vertical extension of the system is finite or whether the flow is virtually unlimited. For the former (a situation typical for river flow), the dispersion coefficient is proportional to (vx )2 ... 

Explain the difference between the dispersion coefficient, dis, in a river and in the atmosphere. How is dis related to the mean advection velocity and to lateral turbulent diffusivity in each case ... 

Figure 24.4 Mixing processes in a river. Ey and E, are the turbulent diffusion coefficients in the lateral and vertical direction, respectively h0 is the maximum depth. Longitudinal dispersion, djs, results from the variation of velocity in a given cross section of the river. A pollutant added to the river in cross section A-B mixes vertically and laterally into the whole river cross-section.

Second, remembering the picture of the railway tracks we used in Section 22.4 to describe dispersion, we concluded that if the number of streamlines (tracks) is limited, then the dispersion coefficient should be inversely related to lateral diffusivity. Thus, from the concept of lateral mixing time (Eq. 24-38) one expects... 

Yet, this is not the only, and usually not even the most important reason why the concentration measured at a fixed location is asymmetric in time. In many cases chemicals enter the river from outfalls (see Fig. 24.4). Remember that vertical mixing usually occurs over a short distance, whereas lateral mixing may need more time (or distance). As discussed before, it is mostly the lateral mixing (or rather its slowness ) which allows longitudinal dispersion. As long as not all streamlines are occupied , the dispersion coefficient is small and the concentration front steep. 

Figure 24.8 Concentration of Xo estimate the dispersion coefficient Edjs we need the lateral turbulent diffusivity Ey measure t a)3 s a"ion°A (26 Tm (see Ecl- 24 45) which in tum is calculated from the friction velocity, u. Problem downstream of spill) and (b) 24.4 deals with the calculation of Edis. As it turns out, a realistic value which agrees...

Consider a fluid flowing steadily along a uniform pipe as depicted in Fig. 2.13 the fluid will be assumed to have a constant density so that the mean velocity u is constant. Let the fluid be carrying along the pipe a small amount of a tracer which has been injected at some point upstream as a pulse distributed uniformly over the cross-section the concentration C of the tracer is sufficiently small not to affect the density. Because the system is not in a steady state with respect to the tracer distribution, the concentration will vary with both z the position in the pipe and, at any fixed position, with time i.e. C is a function of both z and t but, at any given value of z and t, C is assumed to be uniform across that section of pipe. Consider a material balance on the tracer over an element of the pipe between z and (z + Sz), as shown in Fig. 2.13, in a time interval St. For convenience the pipe will be considered to have unit area of cross-section. The flux of tracer into and out of the element will be written in terms of the dispersion coefficient DL in accordance with equation 2.12. For completeness and for later application to reactors (see Section 2.3.7) the possibility of disappearance of the tracer by chemical reaction is also taken into account through a rate of reaction term 9L... 

Besides the time-averaged and the dynamic characterizations, the correlations between different scales add more complexities to the meso-scale structure. For example, the intensive exchange between meso-scale clusters and dispersed particles will reduce the cross-correlation coefficient and then make it hard to discern a cluster from broth. Such complexity requires more efforts in exploring the dominant mechanisms underlying the correlations, which will be discussed and exemplified by multi-scale CFD in later sections. 

In this chapter generalized mathematical models of three dimensional electrodes are developed. The models describe the coupled potential and concentration distributions in porous or packed bed electrodes. Four dimensionless variables that characterize the systems have been derived from modeling a dimensionless conduction modulus ju, a dimensionless diffusion (or lateral dispersion) modulus 5, a dimensionless transfer coefficient a and a dimensionless limiting current density y. The first three are... 

Note that in the Eqs. (147) and (148), the surface concentration cRs is different from the solution concentration cR in the second derivative on the left and in Eqs. (115) and (116) due to convection. Furthermore A is the lateral dispersion coefficient instead of a lateral diffusion coefficient De in the porous electrode case. 

The first EOF, which is responsible for 46.1% of the total dispersion, represents the most large-scale mode of the Black Sea main pycnocline response to external forcing (Fig. 9a). The intra-annual variability of the corresponding coefficient (curve 1 in Fig. 9d) shows that the maximal positive (negative) salinity anomalies in the central (near-shore) areas of the Black Sea described by this mode are observed in April, when the main pycnocline dome is especially high. An opposite situation is observed a half-year later, in October, when the dome is most low. The near-shore zones of maximal positive values... 

Fig. 4 General solution for the dispersion equation on water at 25 °C. The damping coefficient a vs. the real capillary wave frequency o> , for isopleths of constant dynamic dilation elasticity ed (solid radial curves), and dilational viscosity k (dashed circular curves). The plot was generated for a reference subphase at k = 32431 m 1, ad = 71.97 mN m-1, /i = 0mNsm 1, p = 997.0kgm 3, jj = 0.894mPas and g = 9.80ms 2. The limits correspond to I = Pure Liquid Limit, II = Maximum Velocity Limit for a Purely Elastic Surface Film, III = Maximum Damping Coefficient for the same, IV = Minimum Velocity Limit, V = Surface Film with an Infinite Lateral Modulus and VI = Maximum Damping Coefficient for a Perfectly Viscous Surface Film...

Later, taking into account the volume occupied by dispersion particles of not very diluted dispersions led to a more accurate value of the coefficient of the quadratic term ... 

On the other hand, Settari et al. (50) used a finite-element analysis in examining the consec[uences of both velocity-dependent and constant dispersion coefficients during a two-dimensional displacement. They found that fingers in the concentration distribution developed when the permeability was homogeneous, so long as the dispersion coefficients were sufficiently small. This was apparently the first successful use of truncation and round-off errors to play the roles of physical perturbations in initiating instabilities. Russell (51) later had a similar experience. 

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