Response, to pulse input

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.

Fig ure 8-35. Guassian response to pulse input for small dispersion. 

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

The interchange coefficient, kc, has been estimated from the response to pulse inputs of tracer gas [8,13], Patience and Chaouki [8] used sand in a 0.083-m riser and reported values (their k) that ranged from 0.03 to 0.08 m/sec. The values increased with gas velocity, but with considerable scatter in the data. White and coworkers [13] used sand and FCC catalyst in a 0.09-m riser and found k (their k values of 0.05-0.02 m/sec that decreased with gas velocity. The interchange coefficient kc is based on a unit coreannulus area and can be converted to a volumetric coefficient K for comparison with other models. For kc = 0.03 m/sec, AT 1.2 sec which seems the right order of magnitude based on the values shown in Figure 9.12. Flowever, if kc is independent of diameter, D, the volumetric coefficient K will vary inversely with D if rJR is constant. 

Whilst the above is perfectly adequate for the description of processes observed with continuous-wave (cw) input, proper representation of the optical response to pulsed laser radiation requires one further modification to the theory. It is commonly thought difficult to represent pulses of light using quantum field theory indeed, it is impossible if a number state basis is employed. However by expressing the radiation as a product of coherent states with a definite phase relationship, it is relatively simple to construct a wavepacket to model pulsed laser radiation [39]. The physical basis for this approach is that pulses necessarily have a finite linewidth and therefore in fact entail a large number of radiation modes, so that for the pump radiation, it is appropriate to construct a coherent superposition... 

Figure 29.5 (a) Pulse input (b) response of a continuous system to pulse input. 

Scanned pulsed eddy current. This technique for application of eddy-current technology uses analysis of the peak amplitude and zero crossover of the response to an input pulse to characterize the loss of material. This technology has been shown to measure material loss on the bottom of a top layer, the top of a bottom layer, and the bottom of a bottom layer in two-layer samples. Material loss is displayed according to a color scheme to an accuracy of about 5 percent. A mechanical bond is not necessary, as it is with ultrasonic testing. The instrument and scanner are rugged and portable, using conventional coils and commercial probes. The technique is sensitive to hidden corrosion and provides a quantitative determination of metal loss. 

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. 

Curve fitting using a delta function for the pulse input for a TAP reactor should be limited to the latter % part of the response curve for curves of FWHM < 3 times pulse width, while for curves with FWHM > 4 times pulse width, it is a fair approximation fijr most of the curve. The assumption of a zero concentration at the reactor outlet is not good evrai for a pumping speed of 1,500 Is and broad response curves with FWHM > 1000 ms. 

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

We also see another common definition—bounded input bounded output (BIBO) stability A system is BIBO stable if the output response is bounded for any bounded input. One illustration of this definition is to consider a hypothetical situation with a closed-loop pole at the origin. In such a case, we know that if we apply an impulse input or a rectangular pulse input, the response remains bounded. However, if we apply a step input, which is bounded, the response is a ramp, which has no upper bound. For this reason, we cannot accept any control system that has closed-loop poles lying on the imaginary axis. They must be in the LHP. 1... 

The time variations of the effluent tracer concentration in response to step and pulse inputs and the frequency-response diagram all contain essentially the same information. In principle, any one can be mathematically transformed into the other two. However, since it is easier experimentally to effect a change in input tracer concentration that approximates a step change or an impulse function, and since the measurements associated with sinusoidal variations are much more time consuming and require special equipment, the latter are used much less often in simple reactor studies. Even in the first two cases, one can obtain good experimental results only if the average residence time in the system is relatively long. 

For linear systems the relative response to a pulse input is equal to the derivative of the relative response to a step input. Illustration 11.1 indicates how the response of a reactor network to a pulse input can be used to generate an F(t) curve. 

ILLUSTRATION 11.1 DETERMINATION OF AN F(t) CURVE FROM THE RESPONSE OF A REACTOR TO A PULSE INPUT... 

The responses of this system to ideal step and pulse inputs are shown in Figure 11.3. Because the flow patterns in real tubular reactors will always involve some axial mixing and boundary layer flow near the walls of the vessels, they will distort the response curves for the ideal plug flow reactor. Consequently, the responses of a real tubular reactor to these inputs may look like those shown in Figure 11.3. 

The relative response of a single CSTR to an ideal pulse input may be obtained by taking the time derivative of equation 11.1.13. 

The responses of a single ideal stirred tank reactor to ideal step and pulse inputs are shown in Figure 11.4. Note that any change in the reactor inlet stream shows up immediately at the reactor outlet in these systems. This fact is used to advantage in the design of automatic control systems for stirred tank reactors. 

Response of ideal continuous stirred tank reactor to step and pulse inputs. 

Alternative methods of estimating Q)L are based on the response of the reactor to an ideal pulse input. For example, equation 11.1.39 may be used to calculate the mean residence time and its variance. Levenspiel and Bischoff (9) indicate that for the boundary conditions cited,... 

In Illustration 11.1 we considered the response of an arbitrary reactor to a pulse input and used... 

Now consider the determination of this parameter using the variance of the response to a pulse input. The variance measures the spread of the distribution about the mean. For a continuous distribution it is defined as... 

Use the data of Illustration 11.1 for the response of a reactor network to a pulse input to determine the number of identical stirred tank reactors in series that gives a reasonable fit of the experimental data. Use both the slope and variance methods described above. 

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