DYNAMIC RESPONSE OF HEAT EXCHANGERS
T O FLOW RATE CHANGES M. A. LARSON
F. J. S T E R M O L E A N D
Department of Chemical Engineering, Iowa State University, Ames, Iowa
The dynamic response of a double-pipe steam-water heat exchanger was studied both experimentally and theoretically to develop partial and ordinary differential equation models that adequately describe the system behavior resulting from flow rate changes. Both the transient and frequency responses were studied. The partial differential equation model predicted resonance and was in good agreement with experimental data over the full range checked. The ordinary differential equation gave a good representation of the system for transient upsets and for periodic upsets that occur below the resonance frequency. When the resonance frequency i s relatively high because of a short heat exchanger or high mean fluid velocity, use of the ordinary differential equation for linear analysis seems fully warranted.
Two channel
D. C. Amplifier \(
Heat exchanger
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upsets in the input temperature of one of the fluids involved (3, 4, 6, 9, 70). The response of heat exchangers to flow rate changes has been studied less than the response to temperature upsets: because a n energy balance yields partial or ordinary differential equations with variable coefficients for flow disturbances, whereas constant coefficient equations result from temperature upsets. Heat exchange systems may be readily analyzed dynamically for temperature upsets by consideration of either the partial or ordinary differential equation models and an assumption regarding the temperature dependence of the film coefficients. C n the other hand, upsets in flow rate produce differential equations with variable coefficients and are consequently more difficult to handle. Analog computers are helpful in the treatment of these equations but it is still necessary to make several broad assumptions in order to arrive at a solution. This study presents a specific solution to the dynamics problem resulting from varying flow rates in a steam-\cater heat exchanger.
HIS investigation was directed toward the development of Texperimentally verified response equations which may be used to describe adequately the dynamics of heat transfer in double-pipe steam-water heat exchangers. A steam-water exchanger subjected to transient and periodic variations in water flow rate was studied. No attempt was made to study the closed loop system, either experimentally or theoretically. Response data for transient and periodic flow rate upsets were analyzed to develop partial and ordinary differential equation models which adequately describe the system during unsteady-state conditions. Transient tests were made because in practice upsets to a system are often step changes. Periodic tests were made to see if the system could be treated as a linear system and consequently were analyzed using the standard frequency analysis techniques. Many investigators have analyzed the dynamic response of heat exchangers and continuous mixed tank reactors to
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Linear lever control valve
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I&EC FUNDAMENTALS
Experimental apparatus
a
o Experimentol Equation 5
-
n
0
0
0
0
n
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-
Transient increase in flow rate
-8 0
2
4
6
0
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Time, seconds 60 48c
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4
8
12
16
20
24
28
32
Time, seconds Figure 2. A. 8.
Response to changes in transient flow rate Test 1, range 1.96 to 1.57 pounds per second Test 2, range 1.31 to 0.41 pound per second
Experimental Investigation
A flow diagram of the equipment used is shown in Figure 1. Cold water passed the cold junction of a series thermocouple and then passed through the inner pipe of a concentric doublepipe heat exchanger constructed of 1-inch and 2-inch schedule 40 steel pipe. The exchanger was 17.0 feet long and was mounted horizontally. Condensing steam in the annulus heated the water. Heated water leaving the system passed the hot junctions of a bare series thermocouple and passed through a lever-operated linear regulating valve. A twochannel strip chart Mark I1 Brush oscillograph recorder was used to record the thermopile signal. This signal gave a measure of the increase in temperature of the cold fluid while passing through the exchanger. A high gain d.c. amplifier amplified the thermopile signal to a voltage of sufficient magnitude for direct recording. A linear sliding arm potentiometer connected to ,a 1.5-volt dry cell converted the lever a r m position of the regulating valve to a proportional d.c. signal which was recorded o n the other channel of the oscillo-
graph. Inlet water temperature was measured and held to a constant value. Transient flow changes were made by quickly moving the lever valve from its steady-state position to a new position. Steady-state flow rates were measured a t the two valve positions by weighing. Periodic variations in flow rate were conducted with maximum and minimum flow rates equal to the limiting flow rates used for transient testing. Catheron and Hainsworth ( I ) showed that water has almost no capacity for the storage of flow energy and that flow rate is a direct function of valve position a t frequencies below 1 cycle per second. Periodic variations in flow rate were obtained by pulsing the linear lever valve with a mechanical sine generator, which consisted of a cam driven by two Zeromax variable-speed reducers arranged in series. This system produced a good sine wave a t all frequencies between 0.008 and 1 cycle per second. The simplicity of changing frequencies by turning a small crank o n either speed reducer permitted the taking of data rapidly over a wide frequency range. Amplitude of the generated sine waves was varied by changing the length of the valve lever arm. VOL. 2
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FEBRUARY 1 9 6 3
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Variations in flou rate were essentially linear with changes in valve position. Upstream pressure varied slightly with changes in valve position, but this had no significant effect on experimental results because the incremental ranges over which experimental flow rate variations were investigated lvere small and carefully selected, so that flow rates varied linearly with valve position. Experimental results were obtained over a number of different ranges of flow rate for both transient and periodic flow rate upsets with constant steam temperature. All tests were made in the turbulent region. Test 1 constitutes an 11.2% change from the mean flow rate or a 25% change from the minimum flow rate. The minimum and maximum flow rates were 1.57 and 1.96 pounds per second, respectively. The transient response is shown in Figure 2,A. and the frequency response in Figure 3,A. Test 2 constitutes a 52.3y0 change from the mean flow rate or a 220% change from the minimum flow rate. The minimum and maximum flow rates were 0.41 and 1.31 pounds per second, respectively. The transient response is shown in Figure 2.B, and the frequency response in Figure 3,B. The upper transient response curve in Figure 2,B, represents the response to a step change from the maximum to the minimum flow rate. The lower curves represent the response to a step change from the minimum to the maximum flow rate. The fact that the time constants are different for increases and decreases in flow rate is clearly evident, especially for the large change made in test 2. The difference in the system time constants for increases and decreases in flow rate had an important effect on frequency response results also. Because of this nonlinearity in the system, the phase lags were considerably different at the maximum and minimum flow rate peaks. At the maximum flow point, phase lags were less than a t the minimum point because response to flow rate increases is faster than response to flow rate decreases. Phase lags were calculated by measuring the difference between the maximum flow rate peak and its corresponding temperature response peak and averaging it Xvith the difference bettveen the minimum flow rate and its corresponding water response minimum.
A mathematical description of a distributed parameter steam-water heat exchange system may be obtained from heat balances over the water phase and the metal wall ( 3 ) . However, simultaneous solution of these equations is rather lengthy for steam temperature and water flow rate upsets, as shown by Hempel (7). For thin metal walls the capacitance of the wall may be neglected. If this assumption is made, an over-all heat transfer coefficient may be used and a single energy balance made over the water phase of this system will suffice to describe the dynamics. Writing an energy balance over a differential element of length dx of a heat exchanger gives: 30 dt
=
UPdx
- -
at
Capacitance of the metal \Val1 is negligible.
In the solution of Equation 1 it was further assumed that:
3. Steam temperature was uniform over the length of the exchanger at any time. 4 . Heat capacity and density of water do not change appreciably with temperature. 5. Plug flow conditions exist. For flow rate disturbances Equation 1 must be reduced to a constant coefficient equation in order to apply frequency analysis techniques. In a recent article on a related subject (8),Koppel gives a more general solution of Equation I . While this linear perturbation form of the differential enersy equation employed by the authors has been shoum by Koppel to be in error under some conditions, it is the more easily usable form and is sufficiently accurate for the conditions employed experimentally. In this study the equation was simplified in a manner, following the method of Hempel, such that the effect of changes in water flow rate, steam temperature, and inlet water temperature could be studied simultaneously or individually. Heat transfer coefficient variation in the simplified equation has the effect of increasing or decreasing the amplitude of the forcing function. In constant coefficient equations the amplitude of the forcing function does not affect frequency response. Therefore, Equation 1 was solved for frequency response in the partial equation form, assuming the over-all heat transfer coefficient was constant at a value corresponding to the mean flow rate. An ordinary differential equation model Lvith variable heat transfer coefficient is presented later. Let 8. V: and es be expressed as the sum of a steady-state value and a small perturbation from steady state
e
=
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Substituting these quantities into Equation 1 and dropping the terms involving the product of tlvo perturbations give the total energy equation in terms of steady state and dynamic quantities. .At steady state
Subtracting the steady-state equation from the total equation yields the following dynamic equation, which may be solved for the desired input upset.
Mathematical Analysis
Cpp.4 dx
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Equation I C may now be solved by ordinary methods, using to obtain output the boundary condition a t x = 0, 8 = temperature response.
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where B = UP/C,pA. The assumptions used in deriving this equation are: 1. Axial heat conduction is negligible compared to bulk flOW.
64
l&EC
FUNDAMENTALS
For constant steam and input water temperatures
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Frequency, cycles/min.
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Frequency response for flow rate upsets
Test 1, range 1.96 to 1.57 pounds per second Test 2, range 1.31 to 0.41 pound per second VOL. 2
NO. 1
FEBRUARY
1963
65
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the linearized Equation 3 gives resonance a t the same frequency as Equation 4. Because of the nature of the upset, Equation 4 does not result from linearization procedures. To obtain a simpler and more useful mathematical model for flow rate upsets that is easier to use for systems analysis, Equation 1 was simplified to a n ordinary differential equation by letting
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the rate of change of mean fluid temperature with respect to time is assumed equal to the rate of change of output fluid temperature. At time zero, output temperature is assumed equal to zero; therefore other steady-state temperatures are adjusted to this scale.
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Figure 4. Calculated frequency response for water flow rate and steam temperature upsets a t conditions of test 2
Letting the Laplace operator s = iw and solving Equation
3 for magnitude ratio and phase lag give the theoretical curves shown in Figure 4. Agreement between Equation 3 and experimental data is reasonably good for both amplitude ratio and phase lag, especially for the case for a small perturbation (Figure 3). The distributed parameter property of resonance is predicted very well in all cases. Comparison of these results for this single over-all energy balance equation with the results obtained by Hempel for his two-equation model shows that the single equation is much simpler than the two-equation model and agreement with experimental data is as good. I t is interesting to note in solving Equation 3 and from experimental data that magnitude ratio resonance occurs when Lw/V = 2na where w is the upset frequency in radians per second, L is heat exchanger length, V is average fluid velocity, and n is an integer. As a consequence, a short heat exchanger with a high fluid velocity would have resonance conditions only at a very high upset frequency, whereas a long heat exchanger with a low fluid flow velocity would have resonance conditions a t a very low frequency. This factor could be an important consideration in the design of heat exchangers and their control systems. If Equation 2 is solved for constant flow rate and input water temperature but variable steam temperature, the following transfer function results
(4) Figure 4 compares the frequency response of Equations 4 and 3 for conditions existing in experimental test 2. To solve Equation 4, steam temperature upsets were assumed about the steam temperature existing in test 2 and the flow rate used was the average flow rate used in test 2. This comparison shows very good agreement between the two types of upsets and that l&EC FUNDAMENTALS
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420 LL
3
Frequency, cycles/min.
66
where
and c' and denote mean values. For frequency analysis the equation was used as shown. F a n d were evaluated at the mean velocity. U' was taken as a linear function of V', such that the right-hand side of Equation 5 could be expressed as KV'. Figure 3 sho\vs that agreement with experiment was very good for both tests for all frequencies less than the resonance frequencies. It is evident that this simplified mode is preferable to the partial differential equation model at these low frequencies. For calculation of the transient response curves Equation 5 also applies. However, the average flow rate and heat transfer coefficient that are applicable are the values at the new flow rate, since this is the average value after time zero. Figure 2 , A , which represents test 1, shows that for small increases or decreases in the flow rate the curves are nearly mirror images and average values for V and C' could be used. For large changes, however, the difference in the time constants is apparent (Figure 2,B) and average values would not be appropriate. Equation 5 shows that this should be expected, because the time constant involves widely different for large increases and decreases in flow values of 0and about a mean flow rate.
v
v
Discussion Experimental data showed that a single partial differential equation model of a steam-water heat exchange system adequately represents the system. The distributed parameter properties of the system were predicted and agreement between theoretical and experimental was good over the entire frequency range tested. Resonance was shown always to occur for flow rate or steam temperature frequency upsets when Lu/V = 2nn. Thus, the resonance frequency may be predicted from physical parameters of the system involved. For any average flow rate, steam temperature or fluid flow rate upsets give resonance at the same frequency. An ordinary differential equation model of the system was obtained by simplifying the partial differential equation and was shown to be satisfactory for linear systems analysis, if frequency upsets are not expected above the resonance frequency. The ordinary equation does not predict resonance
but gives excellent agreement with experimental transient and frequency results for frequencies below the resonance frequency. Differences in the response curves for transient increases and decreases in flow rate which were observed experimentally were predicted and shown to be due to the fact that the flow rate and over-all heat transfer coefficient appear in the equation time coiistant. From a practical standpoint the simple model Equation 5 has many advantages over the more complicated model Equation 3. Equation 5 is much easier to handle both analytically and for analog computer siniulation. Equation 5 gives agrrement with experiment as good as and even better in some respects than Equation 3 for frequencies less than the resonance frequencies. T h e discontinuities in the amplitude ratio and phase lag curves-i.e., amplitude goes to zero a t the resonance pointwhich result when Equation 3 is used might prove troublesome. Equation 5 gives average values for amplitude ratio and phase lag near the resonance frequencies which in some cases are less in error than the values predicted by Equation 3. No rnen c lature
A = cross-section area, sq. ft. B = UP/C,pA C, = heat capacity of fluid, B.t.u./lb. ’ F. P = inside perimeter of inner pipe, ft. s = Laplace operator t = time, sec. T = time constant. sec. C‘ = over-all heat transfer coefficient, B.t.u./sec. sq. ft. ’ F. V = bulk velocity. ft./sec. 11. = water flow rate, lb./sec. x = axial distance coordinate in exchanger, ft. 0 = temperature, ’ F. w = angular frequency of sinusoidal oscillation, radians/sec. if not noted as cycles/min. p = density of fluid, lb./cu. ft.
SUBSCRIPTS s
= steam
0 = a t initial time zero
i
= exchanger water inlet
L = exchanger water outlet SUPERSCRIPTS ’ = deviation from initial steady-state value _ -- arithmetic average = transformed variable A
References (1) Catheron, A. R., Hainsworth, B. D., Znd. Eng. Chem. 48,
1042-6 (1956). (2) Ceaglske, N. H., “Automatic Process Control for Chemical Engineers,” pp. 98-108, 161-73, Wiley, New York, 1951. (3) Cohen, W. C., Johnson, E. F., Ind. Eng. Chem. 48, 1031-4 (19 56). (4) Debolt, R. R., “Dynamic Characteristics of a Steam-Water Heat Exchanger,” M. S. thesis, University of California, Berkeley, Calif., 1954. (5) Eckman, D. P., “Automatic Process Control,” pp. 269-98, Wiley, New York, 1951. (6) Fanning, R. J., “Dynamic Heat Transfer Characteristics of a Continuous Agitated Tank Reactor,” Ph.D. thesis, University of Oklahoma, Norman, Okla., 1958. (7) Hempel, Arvid, J . Basic Eng. 83, 244-52 (1961). (8) Koppel, L. B., IND.ENG.CHEM.FUNDAMENTALS 1, 131 (1962). (9) Morris, H. J., “Dynamic Response of Shell and Tube Heat Exchangers to Temperature Disturbances” (mimeo.), Eng. Dept., Research and Eng. Division, Monsanto Chemical Co., St. Louis, 1959. (10) Mozley, J. M., Ind. Eng. Chem. 48, 1035-41 (1956). (11) Stermole, F. J., “Dynamic Response of Heat Exchangers to Flow Rate Changes,” M.S. thesis, Iowa State University, Ames, Iowa, 1961. RECEIVED for review May 3, 1962 ACCEPTED December 3, 1962 \York supported by the Iowa Engineering Experiment Station.
VOL. 2
NO. 1
FEBRUARY 1963
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