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A phenomenon that is often present in flow systems is the transportation Zag. Synonyms for this term are dead time and distance velocity lag. As an example, consider the system shown in Fig. 8.6, in which a liquid flows through an insulated tube of uniform cross-sectional area A and length L at a constant volumetric flow rate q. The density p and the heat capacity C are constant. The tube wall has negligible heat capacity, and the velocity profile is flat (plug flow). The temperature x of the entering fluid varies with time, and it is desired to find the response of the outlet temperature y(t) in terms of a transfer function. As usual, it is assumed that the system is initially at steady state; for this system, it is obvious that the inlet temperature equals the outlet temperature; i.e., xs = Ys (8.41)
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If a step change were made in x(t) at t = 0, the change would not be detected at the end of the tube until r set later, where T is the time required for the entering fluid to pass through the tube. This simple step response is shown in Fig. 8.7~. If the variation in x(t) were some arbitrary function, as shown in Fig. 8.7b, the response y(t) at the end of the pipe would be identical with x(t) but again delayed by r units of time. The transportation lag parameter r is simply the time needed for a particle of fluid to flow from the entrance of the pipe to the exit, and it can be calculated from the expression 7=
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q-z A L -
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volume of pipe volumetric flow rate
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(8.42)
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It can be seen from Fig. 8.7 that the relationship between y(t) and y(t) = x(t - 7)
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(8.43)
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FIGURE 8-6 System with transportation lag.
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LINEAR
OPEN-LOOP
SYSTEMS
x (0 r----CUY (0
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FIGURE 8-7 Response of transportation lag to various inputs.
Subtracting Eq. (8.41) from (8.43) and introducing the deviation variables X = x - x, and Y = y - ys give Y(r) = X(t - 7)
(8.44)
If the Laplace transform of X(t) is X(s), the Laplace transform of X(t - 7) is e- X(s). This result follows from the theorem on translation of a function, which was discussed in Chap. 4. Equation (8.44) becomes Y(s) = e- X(s) or Y(s) --s7 X(s)=e
(8.45)
Therefore, the transfer function of a transportation lag is e --s7. The transportation lag is quite common in the chemical process industries where a fluid is transported through a pipe. We shall see in a later chapter that the presence of a transportation lag in a control system can make it much more difficult to control. In general, such lags should be avoided if possible by placing equipment close together. They can seldom be entirely eliminated.
APPROXIMATION OF TRANSPORT LAG. The transport lag is quite different
from the other transfer functions (first-order, second-order, etc.) that we have discussed in that it is not a rational function (i.e., a ratio of polynomials.) As shown in Chap. 14, a system containing a transport lag cannot be analyzed for stability by the Routh test. The transport lag is also difficult to simulate by computer as explained in Chap. 34. For these reasons, several approximations of transport lag that are useful in control calculations are presented here. One approach to approximating the transport lag is to write e - as l/e and to express the denominator as a Taylor series; the result is 1 e -rs= eTs 1 1 + rs + r*s*/2 + r3s3/3! + . .. 1 e -TS g1 + 7s
Keeping only the first two terms in the denominator gives
(8.46)
HIGHER-ORDER SYSTEMS: SECOND-ORDER AND TRANSPORTATION LAG
This approximation, which is simply a first-order lag, is a crude approximation of a transport lag. An improvement can be made by expressing the transport lag as
e -7s = e-d2 -
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