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PROCESS Code 128 Code Set B Scanner In None Using Barcode Control SDK for Software Control to generate, create, read, scan barcode image in Software applications. Code 128 Code Set A Creator In None Using Barcode printer for Software Control to generate, create Code 128B image in Software applications. APPLICATIONS
Decode Code128 In None Using Barcode decoder for Software Control to read, scan read, scan image in Software applications. Code 128 Code Set A Creator In Visual C# Using Barcode creation for .NET Control to generate, create Code 128B image in .NET framework applications. 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 Code 128 Code Set A Generator In .NET Using Barcode drawer for ASP.NET Control to generate, create Code 128C image in ASP.NET applications. Code 128B Encoder In .NET Framework Using Barcode printer for .NET framework Control to generate, create USS Code 128 image in VS .NET applications. Substituting the positive form* of h into Eq. (21.51) gives
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Creating OneCode In None Using Barcode generation for Software Control to generate, create Intelligent Mail image in Software applications. Make Data Matrix 2d Barcode In Visual Basic .NET Using Barcode drawer for VS .NET Control to generate, create DataMatrix image in .NET applications. From this form, we can write by inspection (21.52) Phase angle = & rad (21.53) Painting European Article Number 13 In VS .NET Using Barcode creation for Reporting Service Control to generate, create EAN13 image in Reporting Service applications. Encode EAN / UCC  14 In Java Using Barcode printer for BIRT reports Control to generate, create GTIN  128 image in BIRT reports applications. 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 distancevelocity 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 lumpedparameter systems. The nonminimum phase behavior is typical of distributedparameter systems. Code 3 Of 9 Drawer In .NET Framework Using Barcode maker for .NET framework Control to generate, create Code 3 of 9 image in .NET framework applications. Bar Code Generation In Visual Studio .NET Using Barcode maker for ASP.NET Control to generate, create bar code image in ASP.NET applications. Ransport Lag as a Distributedparameter System
Encode Code 128 In Java Using Barcode generator for Java Control to generate, create Code 128 Code Set B image in Java applications. EAN13 Drawer In None Using Barcode printer for Office Word Control to generate, create UPC  13 image in Word applications. We can demonstrate that the transport lag (distancevelocity lag) is, in fact, a distributedparameter system as follows: Consider the flow of an incompressible fluid through an insulated pipe of uniform crosssectional 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) ewv)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
cLaLtl
FIGURE 218 Obtaining the transfer function of a transport lag from a lumpedparameter 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 wellstirred tank, then the pipe is equivalent to n noninteracting firstorder systems in series, each having a time constant* T = Llnv (Note that taking each zone to be a wellstirred tank is called lumping of parameters.) The overall transfer function for this lumpedparameter 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 distributedparameter system and a lumpedparameter system and indicates that a transport lag is a distributed system. G(s) Heat Exchanger
As our last example+ of a distributedparameter 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 doublepipe, steamtowater heat exchanger. Condensate .
FIGURE 213
Doublepipe 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 heattransfer 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 = crosssectional area for flow inside pipe A,,, = crosssectional area of metal wall Di = inside diameter of inner pipe D, = outside diameter of inner pipe hi = convective heattransfer coefficient inside pipe h, = heattransfer 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

