Residence time distribution impulse response

It is possible to determine the cumulative residence time distribution function F(t) from either a tracer step-change or a tracer impulse response. From its definition, the properties of F(t) are ... 

To understand the behavior of the fluidized bed, one can determine the average residence time or the residence time distribution (RTD) from the tracer technique. For instance, RTD in fluidized bed dryer is usually carried out by means of the stimulus-response technique, in which an impulse of solids marked with some appropriate tracer is fed to the dryer and its time of elution and concentration measured at the exit of the dryer. The material of the tracer has to be such that it can be detected and does not react with the substrate material, and its form of application and response are well known (Levenspiel, 1972). 

The shape of the response curve for an impulse function f t) is shown in Fig. 3.3b for various residence time distribution functions. Figure 3.4c, shows the response curves F t) for a step function. The limiting cases of an ideal stirred vessel and an ideal tubular reactor are shown in both b and c. To quantify RTD curves, two fundamentally different models are used (a) the so-called one-dimensional dispersion model, primarily used for tubular reactors with low backmixing, and (2) the so-called cell model ( tanks in series model ) which was primarily intended for stirred vessel reactors but is of general validity. 

It should be noted that CARPT experiments in the gas—soHd riser produced for the first time the definitive soHds residence time distribution in the riser itself (Fig. 1.12). Precise monitoring of the time when the tracer particle enters the system across the inlet plane, and the time when it exits across either inlet or exit plane, provides its actual residence time in the riser. Ensemble averaging for several thousand particle visits yields the solids RTD. The first passage time distribution is also readily be obtained. This information cannot be obtained by measuring the response at the top of the riser to an impulse injection of tracer at the bottom. By using CARPT, true descriptions ofsoHds residence time distributions can be obtained in the riser (Bhusarapu et al., 2004, 2006). One task of CFD modelers is to develop codes that can predict the experimental observations of CARPT. Here, it... 

In turbulent flow regime (J Cp>I00), the fluid entering each void is considered to be fully mixed and overall dispersion phenomenon can be well described by a tanks-in-series model, where the residence time in a tank is equated with the residence time of flowing fluid in a void of the length Pidf, Pidfl u. Then the tanks-in-series model gives the residence time distribution of the Poisson type (Aris and Amundson, 1957), which can be approximated by the impulse response of the dispersion model by equating... 

Frequently used macro-PDFs are the so-called Internal Age Distribution, /(a), and related Residence Time Distribution (RTD), E 6), which are closely related to the macro-PDF of the velocity, (v). The characteristics and practical use of the RTD are discussed in more detail in the next sub-sections. RTD methods are commonly based on the response of the reactor to a tracer impulse or step given at the reactor inlet. This implies statistically non-stationary calculations, for which (12.6.1-1) can be extended to... 

Extensions of residence time distributions to systems with multiple inlets and outlets have been described (27-29). If the system contains M inlets and N outlets one can define a conditional density function E. (t) as the normalized tracer impulse response in outlet j to input in inlet i as shown schematically in Figure 2. 

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. 

Impulse (delta) response method The input signal is changed in the form of a delta function. This method is widely used in chemical engineering to investigate the residence time probability density distribution function. 

The multiplication P (n)AT gives the existence probability or the probability to have the fluid element in compartment k in the interval of time defined by nAt and (n + 1)At. In other words, it is the response of compartment k to an impulse signal. For k = N, we can observe that the probability Pjj(n)AT makes it possible for the fluid element to leave the cells assembly in the same interval of time T = nAr and t + At = (n + 1)At. Furthermore, because PN-i(n) gives the distribution of the residence time for our assembly of compartments, then we can conclude that the response to one step impulse can be written as ... 

If the tail is truncated, then the tracer impulse response should be normalized based on the area under the curve to obtain a proper density function that approximately describes the distribution of residence times in regions through which there is active flow. 

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