Step forcing function

Fig. 7.27. Response of first-order system to step forcing function...

The number of vessels was chosen by fitting the experimental transient response to a step forcing function of pure helium. The dispersion coefficient (or number of tanks in series) was then assumed to be equal for all species involved. 

Time constant The time required for the output of a first-order system to change 63.2 per cent of the amount of total response to a step forcing function. 

Capacity Element Now consider the case where the valve in Fig. 8-7 is replaced with a pump. In this case, it is reasonable to assume that the exit flow from the tank is independent of the level in the tank. For such a case, Eq. (8-22) still holds, except that/i no longer depends on hi. For changes in fi, the transfer function relating changes in to changes in is shown in Fig. 8-10. This is an example of a pure capacity process, also called an integrating system. The cross sectional area of the tank is the chemical process equivalent of an electrical capacitor. If the inlet flow is step forced while the outlet is held... 

It, therefore, appears that the equilibrium approximation is a special case of the steady-state approximation, namely, the case i > 2- This may be, but it is possible for the equilibrium approximation to be valid when the steady-state approximation is not. Consider the extreme but real example of an acid-base preequilibrium, which on the time scale of the following slow step is practically instantaneous. Suppose some kind of forcing function were to be applied to c, causing it to undergo large and sudden variations then Cb would follow Ca almost immediately, according to Eq. (3-153). The equilibrium description would be veiy accurate, but the wide variations in Cb would vitiate the steady-state description. There appear to be three classes of practical behavior, as defined by these conditions ... 

A forcing function, whose transform is a constant K is applied to an under-damped second-order system having a time constant of 0.5 min and a damping coefficient of 0.5. Show that the decay ratio for the resulting response is the same as that due to the application of a unit step function to the same system. 

Any type of input-forcing function can be used steps, pulses, or a sequence of positive and negative pulses. Figure 14.9a shows some typical input/output data from a process. The specific example is a heat exchanger in which the manipulated variable is steam flow rate and the output variable is the temperature of the process steam leaving the exchanger. 

The plot shown below is realistic, because the forcing function and the relaxation process are overlapping on a temporal basis. These eonsiderations can be formalized, as deseribed for temporal changes of the eoneentra-tion(s) of the reaeting speeies or in terms of the advancements for eaeh of the individual reaction step(s). Knowledge of the forcing function allows one to obtain corrected chemical relaxation data essential for appro-... 

The first term on the right-hand side of eqn. (11) decays away and, after a time approximately equal to 5t, the second term alone will remain. Note that this is a sine wave of the same frequency as the forcing function, but that its amplitude is reduced and its phase is shifted. This second term is called the frequency response of the system such responses are often characterised by observing how the amplitude ratio and phase lag between the input and output sine waves vary as a function of the input frequency, k. To recover the system RTD from frequency response data is more complex tnan with step or impulse tests, but nonetheless is possible. Gibilaro et al. [22] have described a short-cut route which enables low-order system moments to be determined from frequency response tests, these in turn approximately defining the system transfer function G(s) [see eqn. (A.5), Appendix 1]. From G(s), the RTD can be determined as in eqn. (8). 

This result is, of course, identical to that found previously. Responses of a number of CSTRs to other forcing functions may be found with equal ease. Thus, for example, the step response of the three tanks in series considered above is given by... 

The normal function of any control system is to ensure that the controlled variable attains its desired value as rapidly as possible after a disturbance has occurred, with the minimum of oscillation. Determination of the response of a system to a given forcing function will show what final value the controlled variable will attain and the manner in which it will arrive at that value. This latter is a function of the stability of the response. For example, in considering the response of a second order system to a step change, it can be seen that oscillation increases... 

This effect can be forecast on the basis of the retention time distribution function in continuous tank reactors, which represents the simplest approach to the analysis of reactor dynamics. In its cumulative form, this function represents, for any time t, the fraction of the exit volumetric flow rate characterized by a residence time smaller than t and can be measured experimentally by submitting the reactor to a step forcing input in the entering stream. Whereas for the ideal tank reactor, the following... 

Fig. 6.6. Approximation of the force function f t) by a sequence of step-functions.

The relationship between the transient and stationary approaches to the relaxation times has been considered by Eigen and de Maeyer. For any chemical equilibrium a system of nonhomogeneous differential equations which represent the rates of concentration change may be set up. The complete solution of the system is the sum of two solutions. One of these depends on the initial conditions of the dependent variables and upon the forcing function (the transient solution), while the other depends on the differential equation system and on the forcing function (the forced solution). The latter does not depend on the initial conditions of concentration, etc. The step-function methods for studying chemical relaxation experimentally determine the transient behaviour, while the stationary methods determine the steady-state behaviour. 

Commonly encountered forcing functions (or input variables) in process control are step inputs (positive or negative), pulse functions, impulse functions, and ramp functions (refer to Figure 44). 

For the problem at hand, the response in the time domain is first determined. The forcing function is a step change of magnitude 10. Hence,... 

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