Tracers dispersion coefficient determination

Comparison of Models Only scattered and inconclusive results have been obtained by calculation of the relative performances of the different models as converiers. Both the RTD and the dispersion coefficient require tracer tests for their accurate determination, so neither method can be said to be easier to apply The exception is when one of the cited correlations of Peclet numbers in terms of other groups can be used, although they are rough. The tanks-in-series model, however, provides a mechanism that is readily visualized and is therefore popular. 

Fig. 3.3.7 Time dependence of the axial dispersion coefficients D for water flow determined by NMR horizontal lines indicate the asymptotic values obtained from classical tracer measurements. (a) Water flow in packings of 2 mm glass beads at different flow rates and (b) water flow in catalyst.

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. 

Determination of Dispersion Coefficient from Tracer Clouds... 

Now, if we are determining the dispersion coefficient through the use of a pulse tracer cloud, the boundary conditions are those of a Dirac delta ... 

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... 

Rivers are generally considered as a plug flow reactor with dispersion. Determination of the dispersion coefficient for rivers was covered in Chapter 6, and determination of the gas transfer coefficient is a slight addition to that process. We will be measuring the concentration of two tracers a volatile tracer that is generally a gas (termed a gas tracer, C) and a conservative tracer of concentration (Cc). The transported quantity... 

The samples were analyzed for trace metals and sulfate as well as for three fractions of particulate organic matter (POM) using sequential extraction with cyclohexane (CYC), dichloromethane (DCM) and acetone (ACE). Factor analysis was used to identify the principal types of emission sources and select source tracers. Using the selected source tracers, models were developed of the form POM = a(V) + b(Pb) + - - -, where a and b are regression coefficients determined from ambient data adjusted to constant dispersion conditions. The models for CYC and ACE together, which constitute 90% of the POM, indicate that 40% (3.0 pg/m ) of the mass was associated with oil-burning, 19% (1.4 pg/m ) was from automotive and related sources and 15% (1.1 pg/m ) was associated with soil-like particles. 

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. 

Fig. 2.18. Determination of dispersion coefficient from measurements of C-curves at both inlet and outlet of reaction vessel. Tracer added at any convenient point upstream as non-ideal pulse. Small DlIuL value...

When DJuL is found to be large and the tracer response curve is skewed, as in Fig. 2.23b, but without a significant delay, a continuous stirred-tanks in series model (Section 2.3.2), may be found to be more appropriate. The tracer response curve will then resemble one of those in Fig. 2.8 or Fig. 2.9. The variance a2 of such a curve with a mean of tc is related to the number of tanks / by the expression a2 = t2/i (which can be shown for example by the Laplace transform method 7 from the equations set out in Section 2.3.2). Calculations of the mean and variance of an experimental curve can be used to determine either a dispersion coefficient Dl or a number of tanks i. Thus each of the models can be described as a one parameter model , the parameter being DL in the one case and i in the other. It should be noted that the value of i calculated in this way will not necessarily be integral but this can be accommodated in the more mathematically general form of the tanks-in-series model as described by Nauman and Buffham 7 . 

Several investigators have used radioactive tracer methods to determine diffusion rates. Bangham et al. (32) and Papahadjopoulos and Watkins (33) studied transport rates of radioactive Na+, K+, and Cl" from small particles or vesicles of lamellar liquid crystal to an aqueous solution in which the particles were dispersed. Liquid crystalline phases of several different phospholipids and phospholipid mixtures were used. Because of uncertainties regarding particle geometry and size distribution, diffusion coefficients could not be calculated. Information was obtained, however, showing that the transport rates of K+ and Cl" in a given liquid crystal could differ by as much as a factor of 100. Moreover, relative transport rates of K+ and Cl" were quite different for different phospholipids. The authors considered that ions had to diffuse across platelike micelles to reach the aqueous phase. 

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. 

Dispersion coefficients were determined using salt tracer response curves along with the weighted moments method of analysis applied to equation (14) (details in Appendix I). The circular column was 9.5 cm in diameter and 182.2 cm long. 

Here we use a single parameter to account for the nonideality of our reactor. This parameter is most always evaluated by analyzing the RTD determined from a tracer test. Examples of one-parameter models for a nonideal CSTR include the reactor dead volume V, where no reaction takes place, or the fraction / of fluid bypassing the reactor, thereby exiting unreacted. Examples of one-parameter models for tubular reactors include the tanks-in-series model and the dispersion model. For the tanks-in-series model, the parameter is the number of tanks, n, and for the dispersion model, it is the dispersion coefficient D,. Knowing the parameter values, we then proceed to determine the conversion and/or effluent concentrations for the reactor. 

The total dispersion coefficient is usually determined by measuring the concentration distribution of tracer material steadily back-mixed upstream, or by knowing the impulse response of the column for tracer liquid (see Section IV,A,3). 

Based on tracer and solute experiments (Fig. 6.11) the model parameters are determined step-by-step, beginning with the void fraction, total porosity and the axial dispersion coefficient (Section 6.5.6). All experimental data must be corrected for plant effects (Eq. 6.132). 

The axial dispersion coefficient is determined from the concentration profile of a non-penetrating tracer (Tl). A reasonable approximation for its velocity dependence goes back to van Deemter et al. (1956). The axial dispersion coefficient is the sum of the contributions of eddy diffusion and molecular diffusion (Chapter 2.3.4) ... 

The coefficient representing axial dispersion, E, is measured using a tracer by pulse, sinusoidal, or step-change residence time distribution tests, or by measuring backflow. Sometimes the phenomenon is represented by a cefl model, in which the number of well-mixed cells fits dispersion. The coefficient representing radial dispersion, is determined by measuring the radial spread of a tracer from the centerline toward the wall. 

The dispersion coefficient can be detennined from a pulse tracer experiment. Here, we will use / and a to solve for the dispersion coefficient D, and then the Peclet number, Pe Here the effluent concentration of the reactor is measured as a function of time. From the effluent concentration data, the mean residence time. and variance, o, are calculated, and these values are then used to determine Dg. To show how this is accomplished, we will write... 

Several methods for determining have been described in the literature. If a suitable tracer is available, chromatograms at different flow rates should be measured. If parameter estimation is performed to obtain the axial dispersion coefficient or the factor 2, the complete model (including the plant peripherals and the column) as well as all parameters determined so far have to be taken into account. Another possibility is offered by a moments analysis (Section 6.5.3.1, Equation 6.134), which is based on exploiting the coimection between the axial dispersion coefficient and the second moment (Section 6.5.3.1) ... 

Tracer techniques are commonly used to determine the gas dispersion coefficients in fluidized bed reactors. The tracer concentration measured at the outlet in response to a pulse or step input of the tracer at the inlet can be used to calculate the dispersion coefficient based on the dispersion models in a form similar to Eq. (11), i.e.,... 

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