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To obtain the AR and phase angle requires that the magnitude and argument of the right side of Eq. (21.51) be evaluated. This can be done as follows: First write j in polar form; thus j = ,.Mn from which we get
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Substituting the positive form* of h into Eq. (21.51) gives
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From this form, we can write by inspection (21.52) Phase angle = & rad (21.53)
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From these results, it is seen that the AR approaches zero as o + 03 and the phase angle decreases without limit as o + ~0. Such a system is said to have nonminimum phase lag characteristics. With the exception of the distance-velocity lag, all the systems that have been considered up to now have given a limited value of phase angle as w + ~0. These are called minimum phase systems and always occur for lumped-parameter systems. The nonminimum phase behavior is typical of distributed-parameter systems.
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Ransport Lag as a Distributed-parameter System
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We can demonstrate that the transport lag (distance-velocity lag) is, in fact, a distributed-parameter system as follows: Consider the flow of an incompressible fluid through an insulated pipe of uniform cross-sectional area A and length L, as shown in Fig. 21.8~. The fluid flows at velocity v, and the velocity profile is flat. We know from Chap. 8 that the transfer function relating outlet temperature T, to the inlet temperature Ti is
To(s) e-wv)s C(s)
*Notice that the substitution of -(l + j)/ fi into F.Q. (21.51) leads to a result in which the AR is greater than 1 and the phase angle leads. This is contrary to the response of the physical system and is not admitted as a useful solution.
THeoRfrrKxL ANAWSJS OFCOMFLEX FRoclsEs
V Ti (4 V Ti lb) I I I I I I -TO -To
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FIGURE 21-8 Obtaining the transfer function of a transport lag from a lumped-parameter model.
Let the pipe be divided into n zones as shown in Fig. 2l.M. If each zone of length LJn is considered to be a well-stirred tank, then the pipe is equivalent to n noninteracting first-order systems in series, each having a time constant* T = Llnv (Note that taking each zone to be a well-stirred tank is called lumping of parameters.) The overall transfer function for this lumped-parameter model is therefore
To distribute the parameters, we let the size of the individual lumps go to zero by letting n + m. 1 _ . r!z I (L/v)sln + 1 The thermal capacitance is now distributed over the tube length. It can be shown by use of the calculus that the limit is To(s) which is the transfer function derived previously. This demonstration should provide some initial insight into the relationship between a distributed-parameter system and a lumped-parameter system and indicates that a transport lag is a distributed system.
G(s)
Heat Exchanger
As our last example+ of a distributed-parameter system, we consider the doublepipe heat exchanger shown in Fig. 21.9. The fluid that flows through the inner
*This expression for 7 is equivalent to that appearing in Eq. (9.10). Since the transfer function for flow thmgh a tank was developed in Chap. 9, the analysis will not be repeated hen. +The analysis presented here essentially follows that of W. C. Cohen and E. E Johnson (1956). These authors also present the experimental results of frequency response tests on a double-pipe, steam-to-water heat exchanger.
Condensate .
FIGURE 213
Double-pipe heat exchanger.
pipe at constant velocity v is heated by steam condensing outside the pipe. The temperature of the fluid entering the pipe and the steam temperature vary according to some arbitrary functions of time. The steam temperature varies with time, but not with position in the exchanger. Ihe metal wall separating steam from fluid is assumed to have significant thermal capacity that must be accounted for in the analysis. The heat transfer from the steam to the fluid depends on the heat-transfer coefficient on the steam side (h.) and the convective transfer coefficient on the water side (hi). The resistance of the metal wall is neglected. The goal of the analysis will be to find transfer functions relating the exiting fluid temperature T(L, t) to the entering fluid temperature T(O,t) and the steam temperature T,(f). The following symbols will be used in this analysis: T(x, t) = fluid temperature T&x, t) = wall temperature TV(f) = steam temperature T, = reference temperature for evaluating enthalpy p = density of fluid C = heat capacity of fluid pw = density of metal in wall C, = heat capacity of metal in wall Ai = cross-sectional area for flow inside pipe A,,, = cross-sectional area of metal wall Di = inside diameter of inner pipe D, = outside diameter of inner pipe hi = convective heat-transfer coefficient inside pipe h, = heat-transfer coefficient for condensing steam v = fluid velocity ANALYSIS. We begin the analysis by writing a differential energy balance for the fluid inside the pipe over the volume element of length Ax (see Fig. 21.9). This balance can be stated
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