Response to step input

Flow to a vessel is bypassed with a fraction a and a fraction l-p 0f the volume is stagnant. Find the transfer function and the response to step input, Cfu(t-a). 

Radius of injector tube = E/Ro. Dimensionless radius of injector tube Dimensionless response to step input, defined in Section I... 

Fig. 18. Transient response to step-input change of T0 from 573 to 593 K and x 0 from 0.06 to 0.07, type II conditions.

Figure 3.4a Breadboard filter response to step input.

Figure 7.25 Transient response to step input. Same conditions as in Figure 7.24.

Chapter 14 shows how modeling can propose mechanisms to explain experimentally observed oscillations in the cardiovascular system. A control system characterized by a slow and delayed change in resistance due to smooth muscle activity is presented. Experiments on this model show oscillations in the input impedance frequency spectrum, and flow and pressure transient responses to step inputs consistent with experimental observations. This autoregulation model supports the theory that low-frequency oscillations in heart rate and blood pressure variability spectra (Mayer waves) find their origin in the intrinsic delay of flow regulation. 

FIGURE 4b Dynamic signal behavior possibilities in response to step input the network shown in Fignre 4a yields perfect adaptation behavior. SOURCE Reprinted with permission from Lanffenburger. Cop5iright (2000) National Academy of Sciences, U.S.A. 

Figure 5.23 Input-output process (response to step input disturbance).

Calculation of cumulative exit-age distribution function from response to step input of tracer... 

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.

First-Order Response to an Input Step-Change Disturbance... 

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. 

A fluidized bed reactor has a substantial free space above the main level of the catalyst for purpose of disengaging entrainment. In this region plug flow may be assumed to prevail. An overall appropriate model accordingly will consist of well mixed and bypass zones in parallel followed by a plug flow zone. The fraction of flow in bypass is 1-a and the fraction of vessel volume in plug flow is 2. Find the transfer function and equations for the responses to step and impulse inputs of tracer. 

Figure 3.3. Response of various modes of control to step input (Eckman, Automatic Process Control, Wiley, New York, 1958).

The Derivative mode is sometimes referred to as rate because it applies control action proportional to the rate of change of its input. Most controllers use the process measurement, rather than the error, for this input in order to prevent an exaggerated response to step changes in the setpoint. Also, noise in the process measurement is attenuated by an inherent filter on the Derivative term, which has a time constant 1/8 to 1/10 of the Derivative time. Even with these considerations, process noise is a major deterrent to the use of Derivative mode. 

Figure 5. Possible extremes of transient response to step change of reactant input concentration for second order reaction and selected values of kccT. The flow has the same residence time distribution as two perfectly mixed vessels in sequence. Solid curves complete segregation. Broken...

Figure 10.4 Dimensionless response of first-order lag to step input change.

The response of the overdamped multicapacity system to step input change is S-shaped (i.e., initially changes slowly and then it picks up speed). This is in contrast to a first-order response which has the largest rate of change at the beginning. This sluggishness or delay is also known as transfer lag and is characteristic of multicapacity systems. 

Figure 12.3 Response of time-delayed systems to step input change (a) first order (b) second-order.

Figure 6.15 Response to step and pulse inputs for a nonuniform bed. 

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