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Substituting Eq. (l&z) into IQ. (1.3) and rearranging, we have
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The term ~1 has the dimensions of time and is known as the time constant of the tank. We shall study the significance of the time constant in more detail in Chap. 5. At present, it suffices to note that it is the time required to fill the tank at the flow rate, w. Ti is the inlet temperature, which we have assumed is a function of time. Its normal value is Ti,, and qs is based on this value. Equation (1.5) describes the way in which the tank temperature changes in response to changes in Ti and 4. Suppose that the process is pnxeeding smoothly at steady-state design conditions. At a time arbitrarily called zero, the inlet temperature, which was at TiS, suddenly undergoes a permanent rise of a few degrees to a new value Ti, + ATi, as shown in Fig. 1.2. For mathematical convenience, this disturbance is idealized to
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Inlet temperature versus time.
AN INTRODUCNRY
EXAMPLE
Ti,+ATI t r q 0 TimeFIGURE 1-3 Idealized inlet temperahue versus time.
the form shown in Fig. 1.3. The equation for the function Ti(t) of Fig. 1.3 is
(1.6)
This type of function, known as a step function, is used extensively in the study of transient response because of the simplicity of Eq. (1.6). The justification for use of the step change is that the response of T to this function will not differ significantly from the response to the more realistic disturbance depicted in Fig. 1.2. To determine the response of T to a step change in Ti, it is necessary to substitute Eq. (1.6) into (1.5) and solve the resulting differential equation for T(t). Since the process is at steady state at (and before) time zero, the initial condition is
T(0) = TR
(1.7)
The reader can easily verify (and should do so) that the solution to Eqs. (1.5), (1.6), and (1.7) is
T=TR+
!K,-&i)
+ 1 (1 - ,-wwC+l)rh)
(1.8)
This system response, or tank temperature versus time, to a step change in Ti is shown in Fig. 1.4 for various values of the adjustable control parameter K,. The reader should compare these curves with IQ. (1.8), particularly in respect to the relative positions of the curves at the new steady states. It may be seen that the higher K, is made, the better will be the control, in the sense that the new steady-state value of Twill be closer to TR. At first
2:s -v------w
Time--t
FIGURE l-4 Tank temperature Kc.
versus time for various values of
PROCESS SYSTEMS ANALYSIS AND CONTROL
glance, it would appear desirable to make K, as large as possible, but a little reflection will show that large values of K, are likely to cause other problems. For example, note that we have considered only one type of disturbance ip Ti. Another possible behavior of Ti with time is shown in Fig. 1.5. Here, Ti is fluctuating about its steady-state value. A typical response of T to this type of disturbance in Ti, without control action, is shown in Fig. 1.6. The fluctuations in Ti are delayed and smoothed by the large volume of liquid in the tank, so that T does not fluctuate as much as Ti. Nevertheless, it should be clear from E@. (1.4~) and Fig. 1.6 that a control system with a high value of K, will have a tendency to overadjust. In other words, it will be too sensitive to disturbances that would tend to disappear in time even without control action. This will have the undesirable effect of amplifying the effects of these disturbances and causing excessive wear on the control system. The dilemma may be summarized as follows: In order to obtain accurate control of T, despite permanent changes in Ti, we must make KC larger (see Fig. 1.4). However, as K, is increased, the system becomes oversensitive to spurious fluctuations in Tie (These fluctuations, as depicted in Fig. 1.5, are called noise.) The reader is cautioned that there are additional effects produced by changing K, that have not been discussed here for the sake of brevity, but which may be even more important. This will be one of the major subjects of interest in later chapters. The two effects mentioned PIE sufficient to illustrate the problem.
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