Tracer response functions

The distribution of residence times of reactants or tracers in a flow vessel, the RTD, is a key datum for determining reactor performance, either the expected conversion or the range in which the conversion must fall. In this section it is shown how tracer tests may be used to estabhsh how nearly a particular vessel approaches some standard ideal behavior, or what its efficiency is. The most useful comparisons are with complete mixing and with plug flow. A glossary of special terms is given in Table 23-3, and major relations of tracer response functions are shown in Table 23-4. 

A number of special terms are defined in the Glossary, Table 5.1. Equations for tracer response functions are summarized in Table 5.2. 

The unsteady material balances of tracer tests are represented by linear differential equations with constant coefficients that relate an input function Cj t) to a response function of the form... 

In a time period from t = 0 to t = 6t seconds, a quantity m (g) of a tracer is introduced at the system inlet, and the tracer concentration C(t) (g/1) is measured in the exit from the system. Subject to the above conditions, the residence time density function from the measured tracer response is ... 

Impulse Response and the Differential Distribution. Suppose a small amount of tracer is instantaneously injected at time 1 = 0 into the inlet of a reactor. All the tracer molecules enter together but leave at varying times. The tracer concentration at the outlet is measured and integrated with respect to time. The integral will be finite and proportional to the total quantity of tracer that was injected. The concentration measurement at the reactor outlet is normalized by this integral to obtain the impulse response function. ... 

The use of Equation (15.40) is limited to closed systems like that illustrated in Figure 15.10(a). Measurement problems arise whenever /), > 0 or Dgut > 0. See Figure 15.10(b) and suppose that an impulse is injected into the system at z = 0. If Din > 0, some of the tracer may enter the reactor, then diffuse backward up the inlet stream, and ultimately reenter. If Dgut > 0, some material leaving the reactor will diffuse back into the reactor to exit a second time. These molecules will be counted more than once by the tracer detection probes. The measured response function is not f t) but another function, g i), which has a larger mean ... 

The convolution integral and the Exponential Piston Flow Model (EPM) were used to relate measured tracer concentrations to historical tracer input. The tritium input function is based on tritium concentrations measured monthly since the 1960s near Wellington, New Zealand. CFC and SF6 input functions are based on measured and reconstructed data from southern hemisphere sites. The EPM was applied consistently in this study because statistical justification for selection of some other response function requires a substantial record of time-series tracer data which is not yet available for the majority of NGMP sites, and for those NGMP sites with the required time-series data, the EPM and other response functions yield similar results for groundwater age. 

Tracer impulse response data of a commercial hydrodesulfurizer are given by Sherwood (Course in Process Design, 1963) and are given in the first two columns. Various response functions will be found. 

Tracer impulse input data are given in the first two columns. Find various response functions and make plots of F(t) and A(tr). 

A reaction A = B => C is to be conducted in this vessel with Cb0 - 0. In plug flow at the residence time in this vessel conversion of A "would be 70%. What is the concentration of B at the outlet of this reactor in segregated flow Pertinent functions of the tracer response are... 

The purpose of tracer experiments is to extract information about the system in a chemical reaction engineering context, it is the mixing within the system which is of interest, as represented by the system residence time distribution. Because flow mixing is an inherently linear process, the exact form of the RTD which is recovered from a tracer response experiment should be independent both of the amount of tracer used in the test and also of the particular functional form in which the tracer was... 

We are now in the position of being able to predict the form of tracer response curves from systems for which we have theoretical descriptions. From these curves we can recover the system RTD the details of the processing method required to achieve this will depend on the forcing function in question and Sects. 3.2.1—3.2.5 have considered the most common of these. In addition, we have seen in Example 1 how raw experimental measurements can be processed to give E(f) or E(0) data in the absence of any theoretical model. In this section, we now see how to use theoretical or experimental RTD data to predict the conversion which will be expected when a reaction with known kinetics takes place under steady-state conditions in the system under consideration. 

Field data. In the measurements performed In the field, the tracer response was monitored as counts-per-minute as a function of time. This count rate fR) is proportional to the concentration of the radioactive isotope passing through the exit pipe. R.(t) is then taken as the area under the count-rate curve1 and E(t) = R(t)/fc,(t). 

The Laplace transform may be inverted to provide a tracer response in the time domain. In many cases, the overall transfer function cannot be analytically inverted. Even in this case, moments of the RTD may be derived from the overall transfer function. For instance, if Go and GJare the limits of the first and... 

The function f(t) is straightforwardly related to the experimental tracer response curve for a slug input, the downstream concentration vs. time curve is proportional to the function f(t). Response curves for different modes of input theoretically convey equivalent information, although certain inputs are experimentally convenient to carry out. Thus, redoing tracer experiments for different inputs gives no added information about the flow. 

The objective of this study is to develop an analytical model for a soil column s response to a sinusoidally varying tracer loading function by applying the familiar Laplace transform method in which the convolution integral is used to obtain the inverse transformation. The solution methodology will use Laplace transforms and their inverses that are available in most introductory texts on Laplace transforms to develop both the quasi steady-state and unsteady-state solutions. Applications of the solutions will be listed and explained. 

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