Dispersion coefficients tracer injection

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 ... 

As has been discussed, the usual method of finding the dispersion coefficients is to inject a tracer of some sort into the system. The tracer concentration is then measured downstream, and the dispersion coefficients may be found from an analysis of the concentration data. For these tracer experiments there are no chemical reactions, and so r = 0. Also the source term is given by... 

If a pulse of tracer is injected into a flowing stream, this discontinuity spreads out as it moves with the fluid past a downstream measurement point. For a fixed distance between the injection point and measurement point, the amount of spreading depends on the intensity of dispersion in the system, and this spread can be used to characterize quantitatively the dispersion phenomenon. Levenspiel and Smith (L16) first showed that the variance, or second moment, of the tracer curve conveniently relates this spread to the dispersion coefficient. 

Packed Beds. Data on liquid systems using a steady point source of tracer and measurement of a concentration profile have been obtained by Bernard and Wilhelm (B6), Jacques and Vermeulen (Jl), Latinen (L4), and Prausnitz (P9). Blackwell (B16) used the method of sampling from an annular region with the use of Eq. (62). Hartman et al. (H6) used a bed of ion-exchange resin through which a solution of one kind of ion flowed and another was steadily injected at a point source. After steady state conditions were attained, the flows were stopped and the total amount of injected ion determined. The radial dispersion coefficients can be determined from this information without having to measure detailed concentration profiles. 

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. 

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... 

The above example demonstrates that treatment of the basic data by different numerical methods can produce distinctly different results. The discrepancy between the results in this case is, in part, due to the inadequacy of the data provided the data points are too few in number and their precision is poor. A lesson to be drawn from this example is that tracer experiments set up with the intention of measuring dispersion coefficients accurately need to be very carefully designed. As an alternative to the pulse injection method considered here, it is possible to introduce the tracer as a continuous sinusoidal concentration wave (Fig. 2.2c), the amplitude and frequency of which can be adjusted. Also there is a variety of different ways of numerically treating the data from either pulse or sinusoidal injection so that more weight is given to the most accurate and reliable of the data points. There has been extensive research to determine the best experimental method to adopt in particular circumstances 7 " . 

The exact formulation of the inlet and outlet boundary conditions becomes important only if the dispersion number (DjuL) is large (> 0.01). Fortunately, when DjuL is small (< 0.01) and the C-curve approximates to a normal Gaussian distribution, differences in behaviour between open and closed types of boundary condition are not significant. Also, for small dispersion numbers DjuL it has been shown rather surprisingly that we do not need to have ideal pulse injection in order to obtain dispersion coefficients from C-curves. A tracer pulse of any arbitrary shape is introduced at any convenient point upstream and the concentration measured over a period of time at both inlet and outlet of a reaction vessel whose dispersion characteristics are to be determined, as in Fig. 2.18. The means 7in and fout and the variances and out for each of the C-curves are found. 

There are a number of perturbations of these boundary conditions that can be applied. The dispersion coefficient can take on different values in each of the three regions (z < 0 0 z L, and z > 0) and the tracer can also be injected at some point Zi rather than at the boundary, z = 0. These cases and others can be foimd in the supplementary readings cited at the end of the chapter. We shall consider the case when there is no variation in the dispersion coefficient for all z and an impulse of tracer is injected at z = 0 at t = 0. 

Direct estimates of diapycnal exchange coefficients have been made by Kullenberg (1977) from dispersion measurements of injected dye tracer in the thermoline and halocline of the Arkona Basin and the Bornholm Basin in the Baltic Sea. 

Longitudinal dispersion coefficients can be evaluated by injecting a flat pulse of tracer into the bed so that dC/dr = 0. The values of Dl can be estimated by... 

Bader et al. studied both radial and axial gas dispersion using steady-state helium tracer injection [69]. Radial dispersion coefficients ranged between 25 and 67 cmVs. Axial and radial dispersion coefficients were fit to the exact solution of the following differential equation ... 

A radial gas dispersion coefficient can be obtained from the steady-state tracer method by fitting measured radial profiles of tracer concentration downstream of the injection point to a two-dimensional... 

In all reported pulse tests, the tracer was injected on the axis, with the sampling port downstream on the axis. The tracer concentration then reflects the local response on the axis, not the response from the entire cross section. The dispersion coefficient, obtained in this way is the same as the one-dimensional dispersion coefficient defined by equation (37) only for risers of small diameter because of the strong radial variation of tracer concentrations downstream of the injection point. 

Analytical solutions of equation (54) for pulse injection of solid tracers can be obtained with proper boundary conditions by assuming that is independent of radial and axial location (van Zoonen, 1962 Wei et al., 1995b Patience and Chaouki, 1995). If velocity and solids concentration profiles are nonuniform, equation (54) must be solved numerically (Koenigsdorff and Werther, 1995), and the axial and radial solids dispersion coefficients are then obtained by fitting. 

Liquid-phase mixing in three-phase fluidized beds can be described using the dispersion model. A two-dimensional model considers both radial and axial dispersions. Both axial and radial dispersion coefficients are strong functions of operating conditions such as liquid and gas velocities and properties of liquid and solid phases. Evaluations of liquid-phase dispersion coefficients are based on a tracer injection method and subsequent analysis of the mean and the variance of the system response curves. 

Typically, there are two ways to inject tracers, steady tracer injection and unsteady tracer injection. It has been verified that both methods lead to the same results (Deckwer et al., 1974). For the steady injection method, a tracer is injected at the exit or some other convenient point, and the axial concentration profile is measured upward of the liquid bulk flow. The dispersion coefficients are then evaluated from this profile. With the unsteady injection method, a variable flow of tracer is injected, usually at the contactor inlet, and samples are normally taken at the exit. Electrolyte, dye, and heat are normally applied as the tracer for both methods, and each of them yields identical dispersion coefficients. Based on the assumptions that the velocities and holdups of individual phases are uniform in the radial and axial directions, and the axial and radial dispersion coefficients, E and E, are constant throughout the fluidized bed, the two-dimensional unsteady-state dispersion model is expressed by... 

Vail et al. [68] used a steady state technique involving steady injection of aqueous NaCl solution at a given level of the column, to monitor the axial mixing of the liquid. The fluidized bed contained air, water and 0.87 mm spheres with a density of 2700 kg/m. The concentration profiles of tracer measured below the injection point provided information about axial dispersion coefficients and the characteristic mixing length (ratio of the axial dispersion with respect to the fluid velocity). It was observed that the presence of solids and the increase of the gas and liquid velocities were all factors promoting axial mixing of the liquid phase. 

El Temtamy et al. [69] determined the mixing in the liquid phase in a 5 cm. diameter fluidized bed using a steady state tracer technique similar to the one by Vail et al. [68]. Ammonium chloride solution was injected into the fluidized bed at a given level. When a steady state tracer distribution was attained, samples of the liquid upstream from the point of injection were collected. With this information the axial dispersion coefficients were determined in beds of 0.45 mm, 0.96 mm, 2 mm and 3 mm glass beads. No backmixing was detected for 3mm particles. 

From the effluent concentration profile in a polymer or tracer flood, the total core Peclet number is calculated by fitting the analytic form of the convection-dispersion equation as described above. The most direct experimental comparison between the dispersion appropriate for polymer and for an inert tracer should be done in experiments in which both species are present in the injected pulse of labelled polymer solution. This helps to reduce greatly errors that may arise when separate tracer and polymer experiments are carried out. For example, in the study by Sorbie et al (1987d), the dispersion properties of two different xanthans were examined in consolidated outcrop sandstone cores. In all floods, the inert tracer, Cl, was used, thus allowing the dispersion coefficient of the xanthan and tracer to be measured in the same flood. An example of this is shown for a low-concentration (low-... 

The radial dispersion coefficient of solids in a fluidized bed can be evaluated by the injection of tracer particles at the center of the fluidized bed and monitoring the unsteady-state dispersion of these particles [17J. Assuming instantaneous axial mixing of solids and radial mixing occurring by dispersion, the governing partial differential equation of the model, in cylindrical... 

Radial particle dispersion in CFB risers can be studied by measuring radial concentration profiles of tracer particles injected at a single point upstream of the measurement location (van Zoonen, 1962 Wei et al., 1995b). A two-dimensional dispersion model for fully developed axisymmetric flow with constant dispersion axial and radial coefficients, and D t, gives... 

If the Fickian transport coefficient is known, it is possible to predict the distribution of the tracer at any time and location after it is introduced into the column. At the time of injection of the tracer (f = 0), the concentration is high over a short length of column. At a later time fi, the center of the mass of tracer has moved a distance equivalent to the seepage velocity multiplied by fi, and the mass has a broader Gaussian, or normal, distribution, as defined in Eq. (2.6). For this one-dimensional situation, the solution to the advection-dispersion-reaction equation (Eq. 1.5) gives the concentration of the tracer as a function of time and distance. 

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