Input Pulse

Distributions of Residence Times for Chemicai Reactors Chap. 13 

The quantity E(t) is called the residence-time distribution function. It is the fimction that describes in a quantitative manner how much time different fluid elements have spent in the reactor. 

The volumetric flow rate v is usually constant, so we can define E t) as 

An alternative way of interpreting the residence-time function is in its integral form  

We know that the fi action of all the material that has resided for a time t in the reactor between t = Q and f = is 1 therefore, 

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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.

Pulse Inputs Where the sensor within the measurement device is digital in nature, analog-to-digital conversion can be avoided. For rotational devices, the rotational element can be outfitted with a shaft encoder that generates a known number of pulses per revolution. The digital system can process such inputs in either of the following ways ... 

Turbine flowmeters are probably the most common example where pulse inputs are used. Another example is a watt-hour meter. Basically any measurement device that invowes a rotational element can be interfaced via pulses. 

Serial Interfaces Some very important measurement devices cannot be reasonably interfaced via either analog or pulse inputs. Two examples are the following ... 

Eigure 8-10 shows typieal pulse input and output signals. 

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

The two BCs of the TAP reactor model (1) the reactor inlet BC of the idealization of the pulse input to tiie delta function and (2) the assumption of an infinitely large pumping speed at the reactor outlet BC, are discussed. Gleaves et al. [1] first gave a TAP reactor model for extracting rate parameters, which was extended by Zou et al. [6] and Constales et al. [7]. The reactor equation used here is an equivalent form fi om Wang et al. [8] that is written to be also applicable to reactors with a variable cross-sectional area and diffusivity. The reactor model is based on Knudsen flow in a tube, and the reactor equation is the diffusion equation ... 

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

In this example, we have a stirred-tank with a volume Vj of 4 m3 being operated with an inlet flow rate Q of 0.02 m3/s and which contains an inert species at a concentration Cm of 1 gmol/m3. To test the mixing behavior, we purposely turn the knob which doses in the tracer and jack up its concentration to 6 gmol/m3 (without increasing the total flow rate) for a duration of 10 s. The effect is a rectangular pulse input (Fig. 2.7). 

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... 

Frequency methods can give us the relative stability (the gain and phase margins). In addition, we could construct the Bode plot with experimental data using a sinusoidal or pulse input, i.e., the subsequent design does not need a (theoretical) model. If we do have a model, the data can be used to verify the model. However, there are systems which have more than one crossover frequency on the Bode plot (the magnitude and phase lag do not decrease monotonically with frequency), and it would be hard to judge which is the appropriate one with the Bode plot alone. 

A pulse input in which a relatively small amount of tracer is injected into the feed stream in the shortest possible time. 

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. 

In comparison with a pulse input, the step input has the following advantages ... 

The response to a pulse input of tracer may be monitored continuously or by discrete measurements in which samples are analyzed at successive intervals. If discrete measurements are used, nt0/q0 in equation 19.3-2 is approximated by... 

For a step change, a material-balance criterion, analogous to equation 19.3-2 for a pulse input, is that the steady-state inlet and outlet tracer concentrations must be equal, both before and after the step change. Then, it may be concluded that the response of the system is linear with respect to the tracer, and that there is no loss of tracer because of reaction or adsorption. 

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