Pulse tracer inputs

Figure 3.26. Tanks-in-series response to a pulse tracer input from different number of tanks.

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... 

Since tracer experiments are used to obtain RTD functions, we wish to establish that the response to a pulse-tracer input is related to (r) or E(0). For this purpose, c(t) must be normalized appropriately. We call c(t), in arbitrary units, the nonnormalized response, and define a normalized response C(t) by... 

In equation 19.4-50, cA may be replaced by C(0), the normalized response to a Dirac delta (pulse) tracer input at the vessel outlet (z = 1) the normalizing factor to convert... 

C Curve—The Response to an Instantaneous Pulse Tracer Input... 

With this definition for the initial value of Co will generally be greater than 1. Note that equations (E6.9.1) and (E6.9.5) result in = 0. The solution to a pulse tracer input to a complete mix reactor with a leaky dead zone is thus... 

In the vast majority of experimental studies, the backmixing characteristics of a flowing phase are examined using a -pulse tracer input. For the fixed-bed systems shown in Fig. 3-2, if a perfect pulse input is used, then, as shown by Levenspiel,5 6 the axial dispersion coefficient or the Peclet number can be obtained from the variance of the RTD curve. For example, for a closed system and large extent of dispersion, the variance, it, is related to the Peclet number by the equation... 

Figure 13-8 CSTR response to a pulse tracer input.

Open—Open Vessel Boundary Conditions When a tracer is injected into a packed bed at a location more than two or three particle diameters downstream from the entrance and measured some distance upstream from the exit, the open-open vessel boundary conditions apply. For an open-open system an analytical solution to Equation (14-21) can be obtained for a pulse tracer input. For an open-open system the bormdary conditions at the entrance are... 

The following example will show how we can calculate and interpret E t) from the effluent concentratioas from the response to a pulse tracer input to a real (i.e.. nonideal) reactor. 

The premise for the two-parameter model is that we can u.se a combination of ideal reactors to model the real reactor. For example, consider a packed bed reactor with channeling. Here the response to a pulse tracer input would show two dispersed pulses in the output as shown in Figure 13-10 and Figure 14-1. 

The mean residence time of solute in a flow system is of course only a partial description and corresponds to the first temporal moment [11, 23.6-3, p. 756] of exit tracer concentration for a pulse tracer input. It says nothing about the distribution of exit concentration as a function of time or the effect of the internal flow behavior. However, there are many circumstances where the shape is of no importance, and we begin by identifying some of these. 

The curve which describes the concentration-time function of tracer in the exit stream of any vessel in response to an idealized instantaneous or pulse tracer injection is called the C curve. Such an input is often called a delta-function input. As with the F curve, dimensionless coordinates are chosen. Concentrations are measured in terms of the initial concentration of injected tracer if evenly distributed throughout the... 

When we talk about tracers, we generally mean conservative tracers with no sources or sinks. This is opposed to gas tracers, with gas transfer to the atmosphere, and reactive tracers, with a reaction occurring. Tracer studies typically use a conservative tracer, input to the system in a highly unsteady manner, such as a pulse or a front. The pulse and front are typically a more stringent test of the model than a steady-state process with any variety of reactions. Thus, a model that properly simulates the output concentration curve of a pulse or front is assumed to be sufficient for most real conditions with reactions. 

Inputs Although some arbitrary variation of input concentration with time may be employed, five mathematically simple tracer input signals meet most needs. These are impulse, step, square pulse (started at time a, kept constant for an interval, then reduced to the original value), ramp (increased at a constant rate for a period of interest), and sinusoidal. Sinusoidal inputs are difficult to generate experimentally. 

Sloppy Tracer Inputs It is not alw s possible to inject a tracer pulse cleanly as an input to a system, owing to the fact that it takes a finite time to inject the tracer. When the injection does not approach a perfect pulse input (Figure 14-10), the differences in the variances between the input and output tracer measurements are used to calculate the Peclet number ... 

The major aim of the present chapter is to demonstrate how Markov chains can be applied to determine the behavior of a complicated system with respect to the residence time of fluid elements flowing through it. In other words, to obtain the response of the system to some tracer input, usually in the form of a pulse. 

This additivity property of oj allows us to treat any shape of tracer input, no matter how close the input approaches an ideal pulse, and to extract from it the variance ... 

Bubble columns. Tracers are used in bubble columns and gas-sparged slurry reactors mainly to determine the backmixing parameters of the liquid phase and/or gas-liquid or liquid-solid mass transfer parameters. They can be used for evaluation of holdup along the lines reviewed in the previous Section 6.2.1. However, there are simpler means of evaluating holdup in bubble columns, e.g. monitoring the difference in liquid level with gas and without gas flow. Numerous liquid phase tracer studies of backmixing have been conducted (132-149). Steady-state or continuous tracer inputs (132,134,140,142) as well as transient studies with pulse inputs (136,141,142,146) were used. Salts such as KC Jl or NaCil, sulfuric acid and dyes were employed as tracers. Electroconductivity detectors and spectrophotometers were used for tracer detection. The interpretation of results relied on the axial dispersion model. Various correlations for the dispersion... 

Fig. 7. Residence time distributions where U = velocity, V = reactor volume, t = time, = UtjV, Cj = tracer concentration to initial concentration and Q = reactor volume (a) output responses to step changes (b) output responses to pulse inputs.

Kinds oi Inputs Since a tracer material balance is represented by a linear differential equation, the response to anv one kind of input is derivable from some other known input, either analytically or numerically. Although in practice some arbitrary variation of input concentration with time may be employed, five mathematically simple input signals supply most needs. Impulse and step are defined in the Glossaiy (Table 23-3). Square pulse is changed at time a, kept constant for an interval, then reduced to the original value. Ramp is changed at a constant rate for a period of interest. A sinusoid is a signal that varies sinusoidally with time. Sinusoidal concentrations are not easy to achieve, but such variations of flow rate and temperature are treated in the vast literature of automatic control and may have potential in tracer studies. 

Pulse A type of input in which the concentration of tracer in the input stream is changed suddenly, maintained at a non-zero value for a definite period, then changed to the original value and maintained that way for the period of interest. When the pulse is of constant magnitude it is called a square pulse. 

Solution This solution illustrates a possible definition of the delta function as the limit of an ordinary function. Disturb the reactor with a rectangular tracer pulse of duration At and height A/t so that A units of tracer are injected. The input signal is Cm = 0, t < 0 = A/Af, 0 < t < At ... 

Pulse shapes other than rectangular can be used to obtain the same result. Triangular or Gaussian pulses could be used, for example. The Umit must be taken as the pulse duration becomes infinitesimally short while the amount of injected tracer remains finite. Any of these limits will correspond to a delta function input. 

One method of characterising the residence time distribution is by means of the E-curve or external-age distribution function. This defines the fraction of material in the reactor exit which has spent time between t and t -i- dt in the reactor. The response to a pulse input of tracer in the inlet flow to the reactor gives rise to an outlet response in the form of an E-curve. This is shown below in Fig. 3.20. 

Figure 3.20. E-curve response to pulse input of tracer.

In this example, we have a stirred-tank with a volume Vj of 4 m3 being operated with an inlet flow rate Q of 0.02 m3/s and which contains an inert species at a concentration Cm of 1 gmol/m3. To test the mixing behavior, we purposely turn the knob which doses in the tracer and jack up its concentration to 6 gmol/m3 (without increasing the total flow rate) for a duration of 10 s. The effect is a rectangular pulse input (Fig. 2.7). 

We now want to use an impulse input of equivalent "strength," /.e.. same amount of inert tracer added. The amount of additional tracer in the rectangular pulse is... 

A pulse input in which a relatively small amount of tracer is injected into the feed stream in the shortest possible time. 

---