Longitudinal dispersion coefficient

In turbulent flow, axial mixing is usually described in terms of turbulent diffusion or dispersion coefficients, from which cumulative residence time distribution functions can be computed. Davies (Turbulence Phenomena, Academic, New York, 1972, p. 93), gives Di = l.OlvRe for the longitudinal dispersion coefficient. Levenspiel (Chemical Reaction Engineering, 2d ed., Wiley, New York, 1972, pp. 253-278) discusses the relations among various residence time distribution functions, and the relation between dispersion coefficient and residence time distribution. 

In these expressions, B = ZJd, Nps = dVp/EE, Np r = dVn/Eii, where d = some characteristic length such as dp for packed towers or T for spray towers. Ep and Er are the longitudinal dispersion coefficients, which must ultimately be deter-... 

Longitudinal dispersion coefficients can be readily obtained by injecting a pulse of tracer into the bed in such a way that radial concentration gradients are eliminated, and measuring the change in shape of the pulse as it passes through the bed. Since dC/dr is then zero, equation 4.34 becomes ... 

Vazquez and Calvelo (1983b) presented a model for the prediction of the minimum residence time in a fluidized bed freezer which can then be equated to the required freezing time. The model is defined in terms of a longitudinal dispersion coefficient D, which is a measure of the degree of solids mixing within the bed in the direction of flow (and has the dimensions of a diffusivity, and hence units of m s ), a dimensionless time T... 

De Michelis A. and Calvelo, A., Longitudinal dispersion coefficients for the continuous fluidization of different shaped foods, /. Food Eng., 21 (1994) 331-342. 

The second equality in equation (6.28) is a definition of longitudinal dispersion coefficient, Dl- Taylor (1953) assumed that some of the terms in equation (6.28) would cancel and that longitudinal convective transport would achieve a balance with transverse diffusive transport. He then solved the second equality in equation (6.28), for a fully developed tubular fiow, resulting in the relation... 

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. 

Table 6.3 Relationships for longitudinal dispersion coefficient in pipes and channels developed from theory and experiments (after Fisher et al, 1979)...

Dispersion Coefficients in Groundwater Flow. In a uniform media of particles, the longitudinal dispersion coefficient, Dl, and the transverse dispersion coefficient, Dt, are both functions of the grain diameter and velocity. (In our previous example, Dl... 

Koch and Brady (1985) have characterized transverse dispersion coefficient as a fraction of longitudinal dispersion coefficient ... 

Thus, longitudinal dispersion coefficient is roughly 10 times the value of transverse dispersion coefficient in a uniform media. 

Rivers are close to the perfect environmental flow for describing the flow as plug flow with dispersion. The flow is confined in the transverse and vertical directions, such that a cross-sectional mean velocity and concentration can be easily defined. In addition, there is less variation in rivers than there is, for example, in estuaries or reactors - both of which are also described by the plug flow with dispersion model. For that reason, the numerous tracer tests that have been made in rivers are useful to characterize longitudinal dispersion coefficient for use in untested river reaches. A sampling of the dispersion coefficients at various river reaches that were... 

The question that we need to ask ourselves is whether the longitudinal dispersion can be predicted accurately for these rivers. Equation (6.35), which predicts that >l/(m+/i) = constant, is shown in Table 6.4 to have a large range of constants, probably because of the variations in cross section and morphology seen in natural streams. Fisher (1973) observed that this constant seemed to depend on mean surface width, W, and substituted W for h in the numerator of equation (6.34) to develop the following empirical equation to characterize longitudinal dispersion coefficient in rivers ... 

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... 

EXAMPLE 6.14 Determination of longitudinal dispersion coefficient in a river. 

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) ... 

In this reach, on the day of the spill, the Maipo River carried a discharge of 60 m /s, at a mean depth of 2 m, with a mean cross-sectional area of 60 m2 and a stream slope of 0.002. The longitudinal dispersion coefficient is known to be 20 m2/s. 

As described in equation (6.59), longitudinal dispersion coefficient has a 67% confidence interval that is a factor of 1.7 times the best estimate. If the distribution of multiplicative uncertainty is normal, the 95% confidence interval would be at a factor of 3.4 times the best estimate. The reaeration coefficient has are MME of 1.8 for the Thackston and Krenkel equation (equation (9.7)). Again, if the multiplicative distribution is normal, the MME is 0.4 times the 95% confidence interval. Then the 95% confidence interval is a multiplicative factor of 4.5. 

We will use the means and 95% confidence intervals for both and K2 to determine the time, and distance through s = Ut, when Cpeak = 0.0005 g/m. These conditions are listed in Table E9.2.1, with the times determined through iteration on equation (E9.2.3). Table E9.2.1 shows that the peak value of concentration is no longer sensitive to longitudinal dispersion coefficient after roughly 3 days, because the peak is widely spread. The time when the water treatment plants downstream of the spill could turn on the water intake, however, would likely be sensitive to longitudinal dispersion coefficient. 

As for turbulent diffusion (Eq. 22-29), the longitudinal dispersion coefficient dis (dimension L2T ) is defined by ... 

In Illustrative Example 24.4 we calculate the longitudinal dispersion coefficient Edis for the flow regimes I and II of River G. 

Equation 24-47 gives an empirical relation for the longitudinal dispersion coefficient. Explain qualitatively the role of the different parameters in this expression and why they appear in the nominator or denominator, respectively. 

Axial dispersion. An axial (longitudinal) dispersion coefficient may be defined by analogy with Boussinesq s concept of eddy viscosity ". Thus both molecular diffusion and eddy diffusion due to local turbulence contribute to the overall dispersion coefficient or effective diffusivity in the direction of flow for the bed of solid. The moles of fluid per unit area and unit time an element of length 8z entering by longitudinal diffusion will be - D L (dY/dz)t, where D L is now the dispersion coefficient in the axial direction and has units ML T- (since the concentration gradient has units NM L ). The amount leaving the element will be -D l (dY/dz)2 + S2. The material balance equation will therefore be ... 

Dl is the longitudinal dispersion coefficient for an empty tube homogeneous system, u is the linear velocity, and t is the residence time (L/u). 

Clark, J.F., Schlosser, P, Stute, M., and Simpson, HJ. (1996) SF -He tracer release experiment a new method of determining longitudinal dispersion coefficients in large rivers. Environ. Sci. Technol. 30, 1527-1532. 

It should be noted that although the above relation correlates the backmixing coefficient to the fluid properties, no experimental data on systems other than air or nitrogen and water have been reported in the literature. Kato et al.54 obtained data in 6.6-, 12.2-, and 21.4-cm-i.d. columns with particle sizes ranging from 63 through 177 pm and solid concentrations up to 0.2 g solid per cm- of slurry. They correlated the longitudinal dispersion coefficient of the liquid in the slurry by the following dimensionless relation ... 

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