Tracer perfect pulse

In the case where one injects a perfect pulse of tracer Levenspiel and Smith (8) have shown... 

The function f(R) becomes l/Ro , since injection is uniform over the entire plane. The rate of tracer injection, /, is now a function of time. If we define Give as the concentration of injected tracer if evenly distributed throughout the vessel, for a perfect pulse input, we have... 

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... 

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 ... 

Mixing Tests. All these techniques were used in our mixing studies. Because of the difficulty of injecting a perfect pulse of the tracer, the... 

Setting k = 0, simulate the response to a pulse of tracer for 3 perfectly-stirred tanks in series. Repeat this for various numbers of tanks and plot E versus dimensionless time for these on an overlay graph. 

In studying residence time distribution in a tank-flow electrolyzer, a tracer injected into it is recovered in a mixing tank placed downstream. If both tanks are perfectly stirred, then with a tracer pulse, the mole balance for the tracer can be written as... 

Aris (A8), Bischoff (Bll), and Bischoff and Levenspiel (B14) have utilized a method that does not require a perfect delta-function input. The method involves taking concentration measurements at two points, both within the test section, rather than at only one as was previously done. The remaining sketches in Table II show the systems considered. The variances of the experimental concentration curves at the two points are calculated, and the difference between them found. This difference can be related to the parameter and thus to the dispersion coeflScient. It does not matter where the tracer is injected into the system as long as it is upstream of the two measurement points. The injection may be any type of pulse input, not necessarily a delta function, although this special case is also covered by the method. 

For R = 0 we have the pulse at T at the same height as the pulse injected, for the perfect PFTR. For R = 1 the first pulse containing one-half of the tracer will leave at time r/2, and the rest will recycle one-half of this wiU exit at time r, one-fourth will recycle, etc. Thus we will observe pulses at intervals t/2 with heights The sum of all of the infinite number of pulses must be the... 

Next the reactor system in which the CSTR is preceded by the PFR will be treated. If the pulse of tracer is introduced into the entrance of the plug-flow section, then the same pulse will appear at the entrance of the perfectly mixed section Xp seconds later, meaning that the RTD of the reactor system will be... 

To illustrate how dispersion affects the concentration profile in a tubular reactor we consider the injection of a perfect tracer pulse. Figure 14-3 shows how dispersion causes the pulse to broaden as it moves down the reactor and becomes less concentrated. 

The flow configuration comprising of two perfectly-mixed reactors of volumes Vi and V2 is demonstrated in Fig.4.3-1. A tracer in a form of a pulse input is introduced into reactor 1 and is transferred by the flow Qi into reactor 2 where it is assumed to accumulate. Thus, this reactor is a "dead" or "absorbing" state for the tracer, i.e. C = 0 in Fig.4-1. 

The closed recirculation system shown below comprises of Z perfectly-mixed reactors of not the same volume. If a tracer is introduced in a form of a pulse input into the first reactor, the recorder will measure the tracer as it flows the first time, the second time, and so on. In fact, it measures a tracer which passed through Z reactors, 2Z reactors and so on, i.e. the superposition of all these signals. 

The inlet concentration most often takes the form of either a perfect pitlse input (Dirac delta function), imperfect pulse injection (see Figure 13-4), or a step input-Just as the RTD function (/) can be determined directly from a pulse input, the cumulative distribution Fit) can be determined directly from a step input. We will now analyze a step input in the tracer concentration for a system with a constant volumetric flow rate. Consider a constant rate of tracer addition to a feed that is initiated at time t = 0. Before this time no tracer was added to the feed, Stated symbolically, we have... 

To determine the concentrations Cj leaving a set of perfectly mixed volume compartments in series including loops it is necessary to apply Equ. 6.193 to each compartment in sequence. This results in a set of coupled first-order differential equations that are to be solved including kinetics and stoichiometric coefficients, and assuming initial conditions resembling the experimental procedure in which an ionic tracer pulse is injected above the stirrer at t = 0 (thus c = Co in the compartment just above the stirrer and c = 0 (elsewhere). Ait = oo, the concentration is uniform (c = c ). 

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