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Expanding numerator and denominator in a Taylor series and keeping only terms of first-order give 1 - rsl2 lst-order PadC (8.47) e 1 + rsl2
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This expression is also known as a jrst-order Pad.4 approximation. Another well known approximation for a transport lag is the second-order Pad6 approximation: 1 - rsl2 + r2s2/12 e --7s 2: 1 + TSl2 + 72s2/12 2nd-order Pad6
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(8.48)
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The step responses of the three approximations of transport lag presented here are shown in Fig. 8.8. The step response of e e s is also shown for comparison. Notice that the response for the first-order Pad6 approximation drops to - 1 before rising exponentially toward + 1. The response for the second-order Pad6 approximation jumps to + 1 and then descends to below zero before returning gradually back to +1. Although none of the approximations for e - is very accurate, the approximation for e -+ is more useful when it is multiplied by several first-order or second-order transfer functions. In this case, the other transfer functions filter out the high frequency content of the signals passing through the transport lag with the result that the transport lag approximation, when combined with other transfer functions, provides a satisfactory result in many cases. The accuracy of a transport lag can be evaluated most clearly in terms of frequency response, a topic to be covered later in this book.
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FIGURE 8-8 Step response to approximations of the transport lag e - . (1) U(TS + l), (2) lst-order Pad& (3) 2nd-order Padt, (4) CTS.
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8.1. A step change of magnitude 4 is introduced into a system having the transfer function 10 Y(s) -= s* + 1.6s +4 X(s) Determine (a) Percent overshoot (b) Rise time (c) Maximum value of Y(t) (d) Ultimate value of Y(f) (e) Period of oscillation 8.2. The two-tank system shown in Fig. P8.2 is operating at steady state. At time t = 0, 10 ft3 of water are quickly added to the first tank. Using appropriate figures and equations in the text, determine the maximum deviation in level (feet) in both tanks from the ultimate steady-state values and the time at which each maximum occurs. Data:
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AI = A2 = loft* RI = 0.1 ft/cfm R2 = 0.35ftlcfm
FIGURE PS-2
8.3. The two-tank liquid-level system shown in Fig. P8.3 is operating at steady state when a step change is made in the flow rate to tank 1. The transient response is critically damped, and it takes 1.0 min for the change in level of the second tank to reach 50 percent of the total change. If the ratio of the cross-sectional areas of the tanks is Al/A2 = 2, calculate the ratio RI lR2. Calculate the time constant for each tank. How long does it take for the change in level of the first tank to reach 90 percent of the total change
FIGURE P8-3
HIGIIER-ORDER
SYSTEMS: SECOND-ORDER AND TRANSPORTATION LAG
8.4. A mercury manometer is depicted in Fig. P8.4. Assuming the flow in the manometer to be laminar and the steady-state friction law for drag force in laminar flow to apply at each instant, determine a transfer function between the applied pressure p t and the manometer reading h. It will simplify the calculations if, for inertial terms, the velocity profile is assumed to be flat. From your transfer function, written in standard second-order form, list (a) the steady-state gain, (b) 7, and (c) 5. Comment on these parameters as they are related to the physical nature of the problem. p=o
FIGURE P8-4
8.5. Design a mercury manometer that will measure pressures up to 2 atm absolute and will give responses that are slightly underdamped (that is, 5 = 0.7). 8.6. Verify Eqs. (8.17), (8.19), and (8.20). 8.7. Verify Eqs. (8.24) and (8.25). 8.8. Verify Eq. (8.40). 8.9. If a second-order system is overdamped, it is more difficult to determine the parameters 5 and T experimentally. One method for determining the parameters from a step response has been suggested by R. C. Oldenbourg and H. Sartorius (The Dynamics of Automatic Controls. ASME, p. 78, 1948), as described below. (a) Show that the unit-step response for the overdamped case may be written in the form Q-r erlf 2 S(t) = 1 - rle
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