Residence-time distribution 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.

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. 

This program is designed to simulate the resulting residence time distributions based on a cascade of 1 to N tanks-in-series. Also, simulations with nth-order reaction can be run and the steady-state conversion obtained. A pulse input disturbance of tracer is programmed here, as in example CSTRPULSE, to obtain the residence time distribution E curve and from this the conversion for first order reaction. 

Thus we see that the stimulus-response technique using a step or pulse input function provides a convenient experimental technique for finding the age distribution of the contents and the residence-time distribution of material passing through a closed vessel. 

The extent of gas dispersion can usually be computed from experimentally measured gas residence time distribution. The dual probe detection method followed by least square regression of data in the time domain is effective in eliminating error introduced from the usual pulse technique which could not produce an ideal Delta function input (Wu, 1988). By this method, tracer is injected at a point in the fast bed, and tracer concentration is monitored downstream of the injection point by two sampling probes spaced a given distance apart, which are connected to two individual thermal conductivity cells. The response signal produced by the first probe is taken as the input to the second probe. The difference between the concentration-versus-time curves is used to describe gas mixing. 

The experimental measurement and typical results for different residence time distributions in a continuous reactor are summarized in Fig. 3.4. The same arrangements used for determining the mixing time are appropriate for determining the residence time distribution in a reactor. A signal in the form of a pulse or step function or a periodic function is formed at the input, and the response is measured at the output. 

The concentration readings in Table Ell.l represent a continuous response to a pulse input into a closed vessel which is to be used as a chemical reactor. Calculate the mean residence time of fluid in the vessel t, and tabulate and plot the exit age distribution E. 

A similar analysis for a pulse input would give a response curve of Cpuise vs 0, but this can be obtained more easily by differentiating the J ) curve in Fig. 6-5. From Eq. (6-7), the derivative J B) is proportional to The derivative of the dashed line in Fig. 6-5 will be largest at 0 = 0 and will continually decrease toward zero as 0 increases. Such a distribution curve, given as the dashed line in Fig. 6-6, shows that the most probable [largest J ) d6 residence time is at 0 = 0 for a stirred-tank reactor. 

The residence time disttibution of the flow through a given geometry can be thought of as a probability distribution for the time an element of fluid takes to travel through that geometry. It is often easier to envisage an RTD as the response to a perfect, infinitely narrow pulsed input (a Dirac delta input). 

The mean residence time of solute in a flow system is of course only a partial description and corresponds to the first temporal moment [11, 23.6-3, p. 756] of exit tracer concentration for a pulse tracer input. It says nothing about the distribution of exit concentration as a function of time or the effect of the internal flow behavior. However, there are many circumstances where the shape is of no importance, and we begin by identifying some of these. 

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