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Substituting Eq. (6.36) into (6.33) gives
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h - h, 9 - 40, - ___ Rl (6.37)
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At steady state the flow entering the tank equals the flow leaving the tank; thus (6.38) 40 = 40, Introducing this last equation into Eq. (6.37) gives 4 - 4s (6.39)
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Introducing deviation variables Q = q - qs and H = h - h s into Eq. (6.39) and transforming give H(s) = RI (6.40) 7s + 1 Q(s) where RI = 2hi 2/C T = RIA We see that a transfer function is obtained that is identical in form with that of the linear system, Eq. (6.8). However, in this case, the resistance RI depends on the steady-state conditions around which the process operates. Graphically, the resistance RI is the reciprocal of the slope of the tangent line passing through the point (gosh,) as shown in Fig. 6.6. Furthermore, the linear approximation given by Eq. (6.35) is the equation of the tangent line itself. From the graphical representation, it should be clear that the linear approximation improves as the deviation in h becomes smaller. If one does not have an analytic expression such as h 2 for the nonlinear function, but only a graph of the function, the technique can still be applied by representing the function by the tangent line passing through the point of operation. Whether or not the linearized result is a valid representation depends on the operation of the system. If the level is being maintained by a controller at or close to a fixed level h S, then by the very nature of the control imposed on the system, deviations in level should be small (for good control) and the linearized equation is adequate. On the other hand, if the level should change over a wide range, the linear approximation may be very poor and the system may deviate significantly from the prediction of the linear transfer function. In such cases, it may be necessary to use the more difficult methods of nonlinear analysis, some of which are discussed in Chaps. 31 through 33. We shall extend the discussion of linearization to more complex systems in Chap. 21.
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FIGURE 6-6 Liquid-level system with nonlinear resistance.
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In summary, we have characterized, in an approximate sense, a nonlinear system by a linear transfer function. In general, this technique may be applied to any nonlinearity that can be expressed in a Taylor series (or, equivalently, has a unique slope at the operating point). Since this includes most nonlinearities arising in process control, we have ample justification for studying linear systems in considerable detail.
PROBLEMS
6.1. Derive the transfer function H(s)lQ(s) for the liquid-level system of Fig. P6.1 when (a) The tank level operates about the steady-state value of h, = 1 ft. (b) The tank level operates about the steady-state value of hS = 3 ft. The pump removes water at a constant rate of 10 cfm (cubic feet per minute); this rate is independent of head. The cross-sectional area of the tank is 1.0 ft* and the resistance R is 0.5 ft/cfm.
FIGURE P6-1
6.2. A liquid-level system, such as the one shown in Fig. 6.1, has a cross-sectional area of 3 .O ft* . The valve characteristics are q=8h where q = flow rate cfm h = level above the valve, ft Calculate the time constant for this system if the average operating level is (a) 3 ft (b) 9 ft 6.3. A tank having a cross-sectional area of 2 ft* is operating at steady state with an inlet flow rate of 2.0 cfm. The flow-head characteristics are shown in Fig. P6.3. (a) Find the transfer function H(s)lQ(s). (b) If the flow to the tank increases from 2.0 to 2.2 cfm according to a step change, calculate the level h two minutes after the change occurs.
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