Dynamic Characteristics of Double-Pipe Heat Exchangers WILLIAM C. COHEN AND ERNEST F. JOHNSON Princeton University, Princeton, N. J. The performance of automatically controlled process plants depends on the dynamic interaction of all the components in the control loop. To undertake the intelligent design of the control system necessitates knowing the dynamic characteristics of all the components in the control loop as well as having an understanding of closed-loop dynamic behavior. At present very little is known about the dynamic characteristics of the basic processes and operations in the chemical process industries. This article shows how these characteristics may be determined for a simple distributed system. The dynamic characteristics of process components and of the over-all plant may be indicated by frequency response data or by transient response data. From the control standpoint frequency response data are the more useful. It is generally easier to compute the frequency response characteristics than the transient response characteristics since the frequency response is obtained directly from the transfer function of the process, while the transient response requires the taking of inverse Laplace transforms. Furthermore graphical techniques are available for computing transient response from frequency response, and for distributed parameter systems use of these may provide savings in time over use of the inverse Laplace transformation. In general the appraisal of over-all system behavior is more easily visualized from the step response. On the basis of their frequency response behavior four types of individual process components may be distinguished: 1. Lumped-parameter components, where the phase angle reaches a limiting value with increasing frequency and the magnitude ratio decreases to zero for increasing frequency: Lumped parameter systems are those in which the components and their characteristics may be assumed to act at discrete points in the system. Ordinary differential equations govern their behavior, and control system synthesis techniques have been worked out in great detail especially where the equations have constant coefficients and are linear within the operating region. An example of a system that can be assumed to be lumped is a small thermocouple in a well-stirred bath. 2. Distributed-parameter components where the lag angle (negative phase angle) increases without limit and the magnitude ratio decreases to zero with increasing frequency: Distributed parameter systems are those for which the lumping assumption is invalid. The equations governing their behavior are partial differential equations where the space coordinates as well as time must be considered as independent variables. Such systems have been little studied and it remains to be determined to what extent lumpedparameter techniques can be applied to distributed systems. 3. Distributed-parameter components where the lag angle increases without limit as the frequency increases but the magnitude ratio approaches a limiting value greater than zero for increasing frequency, Examples of distributed systems are double-pipe heat exchangers, packed towers, and tubular flow reactors. 4. Pure dead-time components, such as those exhibiting distance-velocity lags, where the phase angle is unlimited but the magnitude ratio is constant at unity for all frequencies: This is a special case of Type 3 component.
Actual plant characteristics may be approximated by combinations of these basic types of components. The simplest to use are the lumped-parameter components and pure deadtime components. This article presents a study of the dynamic characteristics of double-pipe heat exchangers. Although this type problem has been treated in general terms by Gould (3), Farrington (2), and others (1, 6), it is believed that the present treatment not only offers a useful economy of expression without sacrificing rigor, but as a natural consequence leads to the response of the exchanger when simultaneously forced by steam temperature and inlet water temperature. The phenomenon of resonance observed by DeBolt (1) is readily predicted. Figure 1 shows the arrangement of an experimental heat exchanger. It is constructed of concentric 1- and 2-inch brass pipe, 11.6 feet in length mounted horizontally. Steam condenses in the annulus and heats water flowing through the inner pipe. Cold water with velocity V f flows through the inner pipe, and saturated steam condenses in the jacket. Figure 2 shows the model taken for mathematical analysis. The following simplifying assumptions are made: 1. Liquid water is incompressible, and its specific heat and density are constant. 2. Steam temperatures may vary with time but not from point to point in the exchanger at any instant. 3. Axial heat flow is negligible. 4. Outer pipe dynamics may be neglected. 5. Metal wall expansion is negligible; hence the cross-sectional area for each phase is constant. A complete description of the process would involve equations for the conservation of mass, energy, and momentum, coupled with the equations of state and equations for the rate processes occurring between the phases. These equations must be satisfied simultaneously. Without the simplifying assumptions these equations cannot be solved readily. With them, it is only necessary to consider the simultaneous interaction of the heat balances and the rate equations for heat transfer. Only the simplest of heat transfer laws will be considered here namely, that the rate of heat transfer is proportional to the bulk temperature differences between phases.
Equations 1 and 2 are derived from the heat balances, written on the water phase and the metal wall, respectively, for element dx: Since the steam is assumed to be a saturated condensing vapor, its temperature is specified only as a function of time, For the particular system studied the value of the constants are T 1 = 3.69 sec.; T 12 = 2.65 sec.; and T 22 = 1.05 sec. Following Gould (3), if we consider the exchanger in steady state operation at t = 0, and then consider the variation in temperature from this initial condition at any time, t, Equations 1 and 2 become
The steady-state terms are identically zero, and our initial conditions are that the variation in temperature from the steady state is 0 at t = 0. Hence, applying the Laplace transformation we obtain Equations 5 and 6: The simultaneous solution of Equations 5 and 6 for Θ f, leads to Equation 7 Equation 7 represents the total response of the outlet water temperature at position L to the multiple disturbances of steam temperature and inlet water temperature. It is a relationship between the Laplace transforms of the temperatures. If the inlet water temperature is held constant, Equation 7 reduces to
By definition this is the transfer function between the outlet the water temperature and the steam temperature. Similarly, if the steam temperature is held constant while the inlet water temperature is allowed to vary, This equation gives the transfer function between the outlet water temperature and the inlet mater temperature. Equations 8 and 9 show that, if only one type of disturbance is forcing at a time, the transfer function depends only on the system parameters and not on the type of forcing. The transient response may be obtained by substituting the Laplace transform of a unit step input- i.e., l / s, for the Laplace transform of steam temperature or inlet water temperature in Equations 8 or 9 and performing the inverse Laplace transformation. When the inverse transform is not found in the tables, a complex integration must be performed which in the general case is most difficult. The frequency response, however, is obtained simply by replacing s in the transfer function by jω, where j = 1 and ω is the angular frequency in radians per second of an impressed sine wave. The magnitude and phase of the resulting complex number corresponds to the magnitude ratio and phase angle of the frequency response. Transient response data were obtained experimentally by suddenly changing the pressure of the steam from 5 to 15 pounds per square inch gage through the valve before the exchanger. The outlet water temperature response was measured on a Leeds & Northrup Speedomax self-balancing potentiometer. Frequency response data were obtained by sinusoidal varying the set point on the controller which regulates the steam pressure in the pipeline before the exchanger. The pressure variations actually obtained were read on the pressure gage near the exchanger. Steam temperatures were obtained from the steam tables for corresponding pressures. The phase angles were determined by recording the time interval between the steam pressure peaks and the outlet water temperature peaks. In calculating the magnitude ratio from either experimental data or from the theoretical equations
Figure 3 shows the transient response of the heat exchanger. This behavior can be approximated by a system with a 1-second dead-time component and two non-interacting RC stages. The method of Oldenbourg and Sartorius (4) was used to determine the stage time constants. In this method a tangent line is passed through the inflection point of the response curve to define the distances T A and T C. The projection on the 100% asymptote line of the tangent from the time axis to the 100% line is T A, whereas T C is the projection from the inflection point to the 100% asymptote. These parameters give rise to two time constants for the RC stages of 1 second and 3 seconds, respectively. Thus the heat exchanger may be represented by a pure dead time of 1 second and two non-interacting RC stages having time constants of 1 and 3 seconds. Figures 4 and 5 show the frequency response characteristics of the heat exchanger plotted on Bode type diagrams. In Figure 4, circles represent the experimental data, the solid lines represent the theoretical response calculated by substitution of jω for s in Equation 8, and the dashed lines represent the approximation by a pure dead time of 1 second coupled with two RC stages of 1- and 3-second time constants as determined from the transient response. As is typical of Type 2 components the lag angle increases without limit with increasing frequency, and the magnitude ratio decreases to zero with increasing frequency. The lumped approximation for magnitude ratio (dashed line) fits the data well. On the other
hand the theoretically computed curve lies higher than the data. Part of this difference is due to the fact that the experimental data include the response of the temperature measuring means and the pressure measuring means, and the dynamic behavior of the outer jacket wall, while the theoretical curve indicates the response of the heat exchanger alone approximation curve. In addition to the sources of deviation in the magnitude ratio there was a distance-velocity lag between the exchanger and the thermocouple measuring the outlet temperature. Thus it is to be expected that the lag angle would be greater than that indicated from the response of the exchanger alone exclusive of outer wall dynamics. The computed curve is limited by the initial assumptions made in the derivation, and the experimental data are subject to considerable uncertainty at the higher frequencies. Unfortunately, these data do not extend to sufficiently high frequencies to exhibit the resonance indicated by the behavior near a frequency of 2 radians per second. Both the computed magnitude ratio and the computed phase angle resonate at a frequency of about 2 radians per second. These phenomena have been observed experimentally by DeBolt (1) while working with a multipass shell-and-tube steam-to-water exchanger. This resonance is believed to be characteristic of systems forced in a distributed manner. It occurs when the residence time of a slug of water flowing through the exchanger is of the same order of magnitude as the period of the impressed sine wave. Here the L/V ratio is 3 seconds which is approximately the period of the wave. The equations predict that the magnitude ratio and phase angle will show resonances at higher frequencies also.
The curves showing the phase angle indicate that the dashed line approximation is not in good agreement with the experimental results, nor does it indicate resonance. Again the theoretical curve lies above the experimental data, and the data indicate a more rapid falloff than either the computed curve or the approximation curve. In addition to the sources of deviation in the magnitude ratio there was a distance-velocity lag between the exchanger and the thermocouple measuring the outlet temperature. Thus it is to be expected that the lag angle would be greater than that indicated from the response of the exchanger alone. Figure 5 shows the frequency response characteristics when the steam temperature is held constant and a sinusoidal forcing is given to the inlet water temperature. These curves are obtained by substituting jω for s in Equation 9. The system is of Type 3, where the magnitude ratio lies between limits but the lag angle increases indefinitely with increasing frequency. This case was not studied experimentally. Summary Equations for the dynamic characteristics of a simple distributed system have been derived. The frequency response characteristics have been computed, and it has been
shown that distributed periodic forcing leads to resonance in magnitude ratio and phase angle. Typical methods of approximating plant characteristics by means of time delays and time lags give reasonably good representations but are incapable of predicting the resonance phenomena. In certain instances, therefore, this kind of approximation may be undesirable. The importance of identifying the point of application of a disturbance in a system is revealed by comparing the frequency response characteristics relating outlet and inlet water temperatures with frequency response characteristics relating outlet water and steam temperatures. In this article we have not attempted to use our characterization to design an optimum control system for the operation studied. Present knowledge of closed-loop behavior permits us to make a reasonable system design provided we know the characteristics of our process components. In general the dynamic characteristics of chemical engineering processes are not well known. The immediate and pressing need is to determine these characteristics. This article shows how these characteristics may be determined, The simple approaches used here should be generally applicable to distributed systems of all kinds.