Impulse input

Concenti ation The main special lands are Cg, that of the effluent from a vessel with impulse input of tracer = m/V,., the initial mean concentration resulting from impulse input of magnitude m C, that of the effluent from a vessel with a step input of magnitude Cy. 

Impulse An amount of tracer injected instantaneously into a vessel at time zero. The symbol m5(t — a) represents an impulse of magnitude m injected at time t = a. The effluent concentration resulting from an impulse input is designated Cg. 

Residence time, mean The average time spent by the molecules in a vessel. Mathematically, it is the first moment of the effluent concentration from a vessel with impulse input, or ... 

Initial mean concentration with impulse input. 

FIG. 23-7 Imp ulse and step inputs and responses. Typical, PFR and CSTR. (a) Experiment with impulse input of tracer, (h) Typical behavior area between ordinates at tg and ty equals the fraction of the tracer with residence time in that range, (c) Plug flow behavior all molecules have the same residence time, (d) Completely mixed vessel residence times range between zero and infinity, e) Experiment with step input of tracer initial concentration zero. (/) Typical behavior fraction with ages between and ty equals the difference between the ordinates, h — a. (g) Plug flow behavior zero response until t =t has elapsed, then constant concentration Cy. (h) Completely mixed behavior response begins at once, and ultimately reaches feed concentration. 

With an impulse input of magnitude m or an initial mean concentration C = mlVr, the material balance is... 

FIG. Tracer responses to n-stage continuous stirred tank batteries the Erlang model (a) impulse inputs, (h) step input. 

Dispersion Model An impulse input to a stream flowing through a vessel may spread axially because of a combination of molecular diffusion and eddy currents that together are called dispersion. Mathematically, the process can be represented by Fick s equation with a dispersion coefficient replacing the diffusion coefficient. The dispersion coefficient is associated with a linear dimension L and a linear velocity in the Peclet number, Pe = uL/D. In plug flow, = 0 and Pe oq and in a CSTR, oa and Pe = 0. 

The response to an impulse input has the shape of a trapezoid with equations... 

These apparently paradoxical phenomena have been explained by Andronov 4 from the physical point of view, a major point being that at the two nonanalytic points impulsive inputs of energy occur which compensate for the continuous dissipation of energy on the analytic arcs of convergent spirals33 (compare with the theory of clocks of Section 6.9). 

Denervation Loss of nerve impulse input, such as by severing a nerve during surgery or nerve impulse blockade farther up the nerve chain. 

We now want to use an impulse input of equivalent "strength," /.e.. same amount of inert tracer added. The amount of additional tracer in the rectangular pulse is... 

With the impulse input, we use the impulse response in Eq. (2-41) instead, and Eq. (2-44) becomes... 

We can show that with a step input, the output is a ramp function. When we have an impulse input, the output will not return to the original steady state value, but accumulates whatever we have added. (Both items are exercises in the Review Problems.)... 

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... 

Instead of spacing out in the Laplace-domain, we can (as we are taught) guess how the process behaves from the pole positions of the transfer function. But wouldn t it be nice if we could actually trace the time profile without having to do the reverse Laplace transform ourselves Especially the response with respect to step and impulse inputs Plots of time domain dynamic calculations are extremely instructive and a useful learning tool.1... 

Response of ideal plug flow reactor and real tubular reactor to step and impulse inputs. 

Step and impulse inputs. These discontinuous functions are used particularly in Chapter 5. Their definitions and transforms are ... 

Effluent concentrations resulting from impulse and step inputs are designated by C and Cu respectively. The initial mean concentration resulting from an impulse input of magnitude m into a vessel of volume Vr is C° = m/Vr. The mean residence time is the ratio of the volume to the volumetric flow... 

For any series arrangement of CSTRs the overall transfer function is the product of the individual transfer functions. With an impulse input, m<5(t) =... 

Figure 5. 2. Impulse and step inputs and responses. Typical7 PFR and CSTR. CaD Experiment with impulse input of tracer. CbD Typical behavior area between ordinates at ta and tb equals the fraction of the tracer with residence time in that range. Cc3...

Impulse response derived from the step response. The equations for step and impulse inputs may be written in operator form, with f(D) = d/dt + Vs f(D)C = V Cfu(t)... 

C° = hypothetical uniform initial concentration in the whole battery after impulse input. First stage material balance is 0 = V Cj + VrldC1/dt, with C = n 0 yhen t = 0 The transform is,... 

Known properties of the delta function were used in this reduction. Clearly, the impulse input is delayed at the outlet by the period of the residence time. 

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