Dirac pulse

The characterization is performed by means of residence time distribution (RTD) investigation [23]. Typically, holdup is low, and therefore the mean residence time is expected to be relatively short Consequently, it is required to shorten the distance between the pulse injection and the reactor inlet. Besides, it is necessary to use specific experimental techniques with fast time response. Since it is rather difficult, in practice, to perfectly perform a Dirac pulse, a signal deconvolution between inlet and outlet signals is always required. 

Each equivalent site i of a given crystal has the same probability I of being occupied by an electronically excited molecule, immediately after irradiation with a Dirac pulse. The excitation probability of site i is the th element of a vector P that we call excitation distribution among the sites. We distinguish between the low intensity case in which at maximum one dye molecule per crystal is in an electronically excited state and cases where two or more molecules in a crystal are in the excited state. Where not explicitly mentioned we refer to the low-intensity case. 

Temporal analysis of products (TAP) reactor systems enable fast transient experiments in the millisecond time regime and include mass spectrometer sampling ability. In a typical TAP experiment, sharp pulses shorter than 2 milliseconds, e.g. a Dirac Pulse, are used to study reactions of a catalyst in its working state and elucidate information on surface reactions. The TAP set-up uses quadrupole mass spectrometers without a separation capillary to provide fast quantitative analysis of the effluent. TAP experiments are considered the link between high vacuum molecular beam investigations and atmospheric pressure packed bed kinetic studies. The TAP reactor was developed by John T. Gleaves and co-workers at Monsanto in the mid 1980 s. The first version had the entire system under vacuum conditions and a schematic is shown in Fig. 3. The first review of TAP reactors systems was published in 1988. 

Table 1 lists the characteristics of the measured RTD for five different conditions. The first one is shown in Figure 2. The evolution of this curve can be explained by equation (1), although the peaks are not ideal Dirac pulses, because the flow inside the reactor (i.e. the reactor tube (c) and the recirculation pipe (d) in Figure 1) is not of the ideal plug flow type. Therefore, the tracer pulse broadens and eventually spreads throughout the reactor. Nevertheless, the distance between two peaks is a reasonably accurate estimate of the circulation time r/(R+1) in the reactor, and from this the flow through the reactor can be calculated. The recycle ratio R is calculated from the mean residence time r and the circulation time r/(R+l). 

Combining Eqs. 6.43, 6.47 and 6.48 leads directly to the basic equation of chromatography, already presented in Chapter 2 (Eq. 2.16), for the injection of an ideal (Dirac) pulse. 

Contributions to the moments from all parts of the chromatographic plant (Section 6.3.1) are additive in linear chromatography (Ashley and Reilley, 1965). Assuming the hypothetical injection of a Dirac pulse prior to the injector (the injector. .transforms" this into the rectangular pulse of Fig. 6.11) the model parameters can be extracted from the measurements. For plant peripherals, equations for the resulting moments can be found in standard chemical engineering textbooks (e.g. Levens-piel, 1999 and Baerns et al., 1999). 

Figure 2.3a illustrates the typical condition used most often in the theory of linear chromatography (Chapter 6), for the sake of convenience — the Dirac pulse injection or 6 fimction. Although it may correspond to a finite amotmt injected, its width is 0, so the concentration of even a small size injection is imrealistically... 

Equation 2.38 stems from the relationship giving the variance (in length units), of, of a Gaussian peak obtained in chromatography, imder linear conditions, for a Dirac pulse injection (Figure 2.3a) ... 

The BTCs for Dirac pulse injection for Task 6B are shown in Figure 3. Two BTCs are presented for each tracer. The thick lines are for the case when temperature T = 15°C, and the thin lines are for T = 60°C. Clearly the higher the temperature is, the stronger the retention. The tracers can also be divided into three groups as discussed in Task 6A. 

Band broadening effects such as dispersion and mass transfer resistance are represented by the number of tanks (or stages) N. This can be explained by evaluating the moments of the analytical solution of Equation 6.98. For linear isotherms and the injection of an ideal Dirac pulse of one component, this equation yields a gamma density function for the concentration profile. With the retention time Ir lin.i of Equation 6.49 one obtains the elution profile as the time dependence of the concentration in the last tank (k= N) ... 

For a general rate model including axial dispersion mass transfer (Equation 6.80), (apparent) pore diffusion (Equations 6.82, 6.84 and 6.85), and linear adsorption kinetics (Equations 6.32 and 6.33), Kucera (1965) derived the moments by Laplace transformation, assuming the injection of an ideal Dirac pulse. If axial dispersion is not too strong Pe S>4), the equations for the first and second moments can be simplified to (Ma, Whitley, and Wang, 1996)... 

The basic feature of time domain spectroscopy (TDS) is the application of a broad bandwidth signal containing all the frequencies togeflier. A typical signal with broad bandwidth is the square wave (Teorell, 1946). Other popular signals are the Dirac pulse, the multisinusoidal excitation, and the Gauss burst or wavelets. 

The total amount of a nonreactive tracer injected as a Dirac pulse at the reactor entrance is given by q. The Bodenstein number, Ho, is defined as the ratio between the axial dispersion time, = L /D, and the mean residence time, t = r = L ju, which is identical to the space time for reaction mixtures with constant density. For Bo - 0 the axial dispersion time is short compared to the mean residence time resulting in complete backmixing in the reactor. For Ho oo no dispersion occurs. In practice, axial dispersion can be neglected for Ho > 100. 

The only difference is a sudden change in temperature and reaction mass at each injection point J, which can be described by using a Dirac pulse S z) and the Heaviside function ff (z). Reactant A2 contained in flow 2 is injected in deficit to reactant Ai into flow 1 before reaching the last point, where the stoichiometric balance is attained. It is assumed that the volume of the injected reactant is equal at each injection point and that it mixes instantaneously with the main stream. 

Linear velocity of gas bubble (slug), of liquid phase, mean velocity, of cooling medium/fluid, of reaction mixture Dirac pulse... 

Fig.3. Computed oxygen concentration at 3 ms, assuming a Dirac pulse at time 0, or an exponential release (time-constant 1 ms). The curves were normalized at the origin.

According to linear response theory, the procedure described is valid independent of the particular type of perturbation, e.g., sinusoidal, multi-sinus, step function, Dirac pulse, white noise, etc., provided the system meets the following conditions (Kramers, 1929 de Kronig, 1926 Van Meirhaeghe et al., 1976 Macdonald and Urquidi-Mac-donald, 1985 Urquidi-Macdonald et al., 1986). 

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