Tracers dispersion model

At a close level of scrutiny, real systems behave differently than predicted by the axial dispersion model but the model is useful for many purposes. Values for Pe can be determined experimentally using transient experiments with nonreac-tive tracers. See Chapter 15. A correlation for D that combines experimental and theoretical results is shown in Figure 9.6. The dimensionless number, udt/D, depends on the Reynolds number and on molecular diffusivity as measured by the Schmidt number, Sc = but the dependence on Sc is weak for... 

Washout experiments can be used to measure the residence time distribution in continuous-flow systems. A good step change must be made at the reactor inlet. The concentration of tracer molecules leaving the system must be accurately measured at the outlet. If the tracer has a background concentration, it is subtracted from the experimental measurements. The flow properties of the tracer molecules must be similar to those of the reactant molecules. It is usually possible to meet these requirements in practice. The major theoretical requirement is that the inlet and outlet streams have unidirectional flows so that molecules that once enter the system stay in until they exit, never to return. Systems with unidirectional inlet and outlet streams are closed in the sense of the axial dispersion model i.e., Di = D ut = 0- See Sections 9.3.1 and 15.2.2. Most systems of chemical engineering importance are closed to a reasonable approximation. 

Given k fit) for nny reactor, you automatically have an expression for the fraction unreacted for a first-order reaction with rate constant k. Alternatively, given ttoutik), you also know the Laplace transform of the differential distribution of residence time (e.g., k[f(t)] = exp(—t/t) for a PER). This fact resolves what was long a mystery in chemical engineering science. What is f i) for an open system governed by the axial dispersion model Chapter 9 shows that the conversion in an open system is identical to that of a closed system. Thus, the residence time distributions must be the same. It cannot be directly measured in an open system because time spent outside the system boundaries does not count as residence but does affect the tracer measurements. 

The response of the axial dispersion model to step or pulse tracer inputs can be determined by writing a material balance over a short tubular segment and then solving the resultant differential equations. A transient material balance on a cylindrical element of length AZ gives... 

Effluent response curves for perfect impulse injection of tracer (axial dispersion model). (Adapted from Chemical Reaction Engineering, Second Edition, by O. Levenspiel. Copyright 1972. Reprinted by permission of John Wiley and Sons, Inc.)... 

In Section 11.1.3.1 we considered the longitudinal dispersion model for flow in tubular reactors and indicated how one may employ tracer measurements to determine the magnitude of the dispersion parameter used in the model. In this section we will consider the problem of determining the conversion that will be attained when the model reactor operates at steady state. We will proceed by writing a material balance on a reactant species A using a tubular reactor. The mass balance over a reactor element of length AZ becomes ... 

Determining Pe, from Tracer Data As noted in Section 19.4.2.2.1, values of PeL, the single parameter in the axial dispersion model, may be obtained from the characteristics of the pulse-tracer response curve, C(0) = E(6). 

The aim of dispersion models is the prediction of atmospheric dilution of pollutants in order to prevent or avoid nuisance. Established dispersion models, designed for the large scale of industrial air pollution have to be modified to the small scale of agricultural pollutions. An experimental setup is described to measure atmospheric dilution of tracer gas under agricultural conditions. The experimental results deliver the data base to identify the parameters of the models, For undisturbed airflow modified Gaussian models are applicable. For the consideration of obstacles more sophisticated models are necessary,... 

The published guideline VDI 3881 /2—4/ describes, how to measure odour emissions for application in dispersion models. Results obtained by this method have to be completed with physical data like flow rates etc. As olfactometric odour threshold determination is rather expensive, it is supplemented with tracer gas emissions, easy to quantify. In the mobile tracer gas emission source, fig, 2, up to 50 kg propane per hour are diluted with up to 1000 m2 3 air per hour. This blend is blown into the open atmosphere. The dilution device, including the fan, can be seperated from the trailer and mounted at any place, e.g. 

Fortunately, it is not always necessary to recover the system RTD curve from the impulse response, so the complications alluded to above are often of theoretical rather than practical concern. In addition, the dispersion model is most appropriately used to describe small extents of dispersion, i.e. minor deviations from plug flow. In this case, particularly if the inlet pipe is of small diameter compared with the reactor itself, the vessel can be satisfactorily assumed to possess closed boundaries [62]. An impulse of tracer will enter the system and broaden as it passes along the reactor so that the observed response at the outlet will be an RTD and will be a symmetrical pulse, the width of which is a function of DjuL alone. 

Figure 13.1 The spreading of tracer according to the dispersion model.

On the assumption that the closed vessel of Example 11.1, Chapter 11, is well represented by the dispersion model, calculate the vessel dispersion number D/uL. The C versus t tracer response of this vessel is... 

In the preceding section we discussed the dispersion model which can account for small deviations from plug flow. It happens that a series of perfectly mixed tanks (backmix flow) will give tracer response curves that are somewhat similar in shape to those found from the dispersion model. Thus, either type of model could be used to correlate experimental tracer data. 

The discussion in Section III,C showed that there was no unique way to compare the stirred tank and dispersion models based on the tracer curves. Each different basis of comparison gave different results. The two models have been compared for chemical reactions by van Krevelen (V6), Trambouze (TIO), and Levenspiel (L13a). Levenspiel used Figs. 28 and 29 to determine the correspondence of the models. His results are shown in Fig. 30. The various criteria give results that differ increasingly with rise in reaction order, conversion, and degree of mixing. 

Dimensionless response curve to a pulse input, defined in Section I Concentration Initial concentration of tracer or reactant entering the vessel or reactor Average concentration of tracer in system Integral average tracer concentration in vessel during steady state injection (dispersion model)... 

The assumption that Cou = 1 in equation (6.43) is really only accurate when Pe > 10. The only way to apply this tracer curve to the plug flow with dispersion model while Cou 1 would be to route each portion of the tracer curve through the reactor. With Pe = 9.4, this solution will be close, although stiU an approximation. 

The plug flow with dispersion model results in a degradation to 3.4% of the inflow trichloroethylene concentration. This is significantly different than the plug flow model (1.0%). It is also a more accurate solution. Whether it is the tail of a tracer pulse or a reaction that approaches complete degradation, one needs to be careful about applying the plug flow model when low concentrations, relative to the inflow, are important. 

Both the tanks-in-series and the plug flow with dispersion models have an effective dispersion that we have related to the variance of the tracer cloud ... 

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... 

Public concern about industrial chemical exposures might also be misguided. The EPA typically uses mathematical dispersion models to calculate human exposure to chemicals released into the air by major stationary sources like factories and power plants. There is little evidence that the models are predictive. In one experiment, a tracer gas was released from the Alaska pipeline terminus at Valdez. Actual exposure, as measured by personal exposure badges, was compared with the predictions of the EPA dispersion model. The statistical correlation between them was near zero (— 0.01), meaning the predictions were worthless (Wallace 1993, 137-38). 

Mass transport in laminar flow is dominated by diffusion and by the laminar velocity profile. This combined effect is known as dispersion and the underlying model for the theoretical derivation of a kinetic study had to be derived from the dispersion model, which Taylor [91] and Aris [92] developed. Taylor concluded that in laminar flow the speed of an inert tracer impulse initially given to a channel will have the same speed as the steady laminar carrier gas flow originally prevailing in this channel. 

A reactive dispersion model proposed by Muller et al. [37] predicts a change of speed if the tracer impulse consists of reactants, which react at the walls of the channel (Figs. 4.29 and 4.30). Brenner found a quite fascinating explanation for... 

Both the tank in series (TIS) and the dispersion plug flow (DPF) models require tracer tests for their accurate determination. However, the TIS model is relatively simple mathematically and thus can be used with any kinetics. Also, it can be extended to any configuration of compartments with or without recycle. The DPF axial dispersion model is complex and therefore gives significantly different results for different choices of boundary conditions. 

Methods for evaluating the axial dispersion coefficient from RTD data As mentioned earlier, the one-parameter axial-dispersion model is widely used to correlate RTD data. The nature of the RTD depends upon the nature of the tracer input and the nature of the. flow, characteristics. For the RTD shown in Fig. 3-4 o), the axial dispersion coefficients for the liquid and solid phases can be obtained by fitting the equation... 

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