Tracer Dirac delta

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... 

Since all tracer entered the system at the same time, t = 0, the response gives the distribution or range of residence times the tracer has spent in the system. Thus, by definition, eqn. (8) is the RTD of the tracer because the tracer behaves identically to the process fluid, it is also the system RTD. This was depicted previously in Fig. 3. Furthermore, eqn. (8) is general in that it shows that the inverse of a system transfer function is equal to the RTD of that system. To create a pulse of tracer which approximates to a dirac delta function may be difficult to achieve in practice, but the simplicity of the test and ease of interpreting results is a strong incentive for using impulse response testing methods. 

X/i o. Dimensionless axial variable = X/L. Dimensionless axial variable Dirac delta function, see reference (S20) Fraction voids in packed bed Eigenvalue in Eq. (67c), (68c), and (69c) Mean of tracer curve at measurement point (dimensionless) Difference in means of the tracer curves at the two measurement points Xm and Xa Kinematic viscosity of fluid... 

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 ... 

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... 

The mathematical statement of the above type of experiment is to inject an impulse of tracer into the vessel inlet at time zero. This is represented by the Dirac delta function or perfect unit impulse function ... 

The entire amount of the tracer is fed to the reactor inlet within a very short time to approach the Dirac delta function as close as possible. The Dirac function has the following properties ... 

For ease of interpretation, it is important that the input tracer signal should be an ideal Dirac delta function. This is difficult to achieve manually, especially when distortion by peripheral equipment is taken into consideration. 

RTD methods are based on the concept of age distribution functions and make use of the experimentally measured or calculated residence time distribution of fluid elements in a reactor vessel (Figure 12.3-1, C and D). A Lagrangian perspective is taken and the age of a fluid element is defined as the time elapsed since it entered the reactor. In what follows, steady state operation of a vessel fed with a volumetric flow rate F is considered. A residence time distribution (RTD) experiment can be performed with inert tracers, such that at an instant of time all fluid elements entering a reactor or process vessel are marked. The injection of an impulse of tracer into the vessel at time zero can be mathematically represented by means of the Dirac delta function or perfect unit impulse function ... 

The pulse of tracer that was injected can be described by means of the Dirac delta function, 8 t). The properties of the delta function are... 

The Dirac delta function provides a mathematical description of the sharp pulses of tracer that were shown in the preceding figures. 

The derivation of Eqn. (10-2) was designed to illustrate the use and properties of the Dirac delta function. Of course, Eqn. (10-2)is valid for any kind of tracer injection where the amount injected is Mq. The tracer does not have to be injected as a sharp pulse, i.e., a Dirac delta function. The amount of tracer that leaves the vessel over all time must be equal to the amount that was injected, independent of the shape of the input function. 

This result agrees with the qualitative analysis that we performed in Section 10.2.2.1. It could have been deduced without going through the formality of solving Eqn. (10-13). In a PFR, each and every element of fluid spends exactly the same time in the reactor. For a constant-density fluid, that time is V/i> = t. Therefore, if we inject a Dirac delta function of tracer at t = 0, a Dirac delta function wiU emerge at t = r. This is a necessary consequence of the fact that there is no mixing in the direction of flow in a PFR, and no gradients in the direction normal to flow. 

Suppose that a pulse of tracer enters the first reactor (the PFR) as a Dirac delta function at r = 0. The tracer will emerge from the PFR and enter the CSTR as a delta function at r = r. Tracer will not appear in the effluent from the CSTR until t = r, since there is no tracer in the CSTR during the time period 0[Pg.396]

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