PROCESS CONTROL
Dynamic Response Analysis of Air Heater Temperature Control System LESLIE M. ZOSS, NORMAN W. GOLLIN,
AND
ROBERT 1. EDELMAN
Taylor Instrument Companies, Rochester, N. Y.
T
HE end purpose or aim of most control studies is economic,
better products for less money and/or reduced original costs. T o the control engineer, improved control means better product for less money. To obtain the optimum in control, the process and control elements must be matched to produce specified stability, recovery time after disturbances, and correction for load changes (IS,19). Modern controllers are provided with a variety of responses t h a t are adjustable so t h a t the control system may be made t o perform as required ( 1 , 11, 18). Frequently, the desired control system performance cannot be obtained by making the standard available adjustments after the system has been installed. Other steps must be taken to improve the quality of control. Dynamic response of a control system is a function of all the 10). Sometimes elements t h a t make up the control loop (9,4,6, the performance characteristics can be improved by proper choice of control elements or by their relocation. These changes may be simple and not very costly t o make. Controllability may also be improved by modifying process equipment. However, changes t h a t often make the most improvement are too costly, and there must be a compromise for
what otherwise could have been a much superior performing control system. If the control system performance could be predicted in the design stage, any changes necessary to good control could be incorporated before the equipment is built and the system assembled, thereby eliminating costly modifications and possible delays in start-up (8, 9). This report describes a technique whereby the dynamic response characteristics of the control system can be calculated from the system constants. T o verify the validity of the technique a n analysis was made on the dry bulb control system of B Kathabar air-conditioning system (17’) at the Bell and Howell plant in Rochester, N. Y . The system (Figure 1) provides air at constant temperature and humidity t o a dryer. The dry bulb air temperature is measured after the filter by a thermal bulb and temperature transmitter. The output actuates the temperature recording receiver controller. A portable pneumatic sine wave generator (16), having a frequency range from 0.0023 to 44 cycles/minute in multiple steps of 1.6, provided the sine wave input. This input was introduced at the valve motor controlling steam flow t o the heater. The
SPRAY CHAMBER
Figure 1. June 1956
Air temperature control system
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
1069
ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT output of tlie ir:irisniitttr \\ ; i < u w l :is r!.c y . . t G i i i oiltpu:. 1'iIeum:,ric$ amplifiers, liaving a fist gsin rind phase response \\-ithin the frequency range of the generator, \\-ere used on both the input and output signals, The amplifier outputs were recorded on a t1v-o-pen Transet (Taylor Instrument Companies) recorder. The recorder was driven synchronously with the generator. I n this way all Tvaves were of equal length, independent of the frequency. For the equipment used this length was 6 inches.
114:11.
TEMP E R ATU RE
ym
(1)
(2)
A heat balance on tube walls shon-s that the change of heat flow from the 'Ondensing the rate Of change Of heat storage in the tube walls plus the change of heat flow to the air:
The resistance of the steam film \vas assumed t o be negligible so the temperature of the tube walls equals the steam temperature. The temperature of the air was taken as the average between the air inlet and o u t k t temperatures. since the illlet air temperature is constant, the change of average air temperature is half the change of outlet air temperature. A heat balance on the air flowing through the heater requires the change of heat floiv from the tubes t o equal the change of heat flow to the air.
1070
ml QaCa
T[: +
(7)
('"
-
A@/) = W/C,pAO,
(8)
(assuming the wave length is much greater than the filter thickness). The heat balance on the air passing through the filter becomes
(Definitions and numerical values for the various palameters are given in thc Nomenclature.) Since the steam in the jacket is saturated there is a unique relationship between changes in steam pressure and changes in steam temperature, thus =
(6)
hhi4h
[ f i - 2&,c, -
12, 1.4, 1 6 ) at the operating point, the partial deii>atives ( a , p, y ) may be taken as the slopes of a characteristic curve at the operating point. Values of a and p are from manufacturers' literature, and y may be found in the steam tables. The next element in the system is the air filter which is shown in Figure 2 . I n this case the desired relationship is the change in air output teniperature which results from a change in air input temperature. A heat balance on the mass of the filter shoa s that the change of heat flow through the air film equals the change in rate of heat storage in the filter:
hrA/
First consider the dynamics of the air heater shown iri Figure 2. The relationship between a change of air pressure to the valve and the resulting change of outlet air temperature from the heater \vi11 be determined. Assuming a constant pressure steam supply, the flom of steam through the valve depends on the pressure in the steam jacket and the valve position. The valve position in turn is related directly t o the air pressure in the valve motor. The variation of steam flow for small changes is
AP,
a hhAh
If the system is linearized
Schematic process
+ PAP,
fly&oc, hh.4h T'llhchg
I
aAP,
1 =
=
AIR FILM R IS IS TA NC E
=
to
'
ES
a&,
(4,
where
p4 FROM CONTROLLER
Figure 2.
= Cz,C',102
-l
We can obtain the desired relationship for the heater by solving Equations
Analysis of System The dynamic characteristics for each element in this system were derived. The response of the entire system was then determined by combining the responses of each of the elements (3). When analyzing for control purposes i t is necessary to determine the changes of the variables measured from their equilibrium values following an upset.
(10" - 1\28.)
QaCa(A0z - A83) = h,A/
(*"
+ 2
- AOf)
(9)
Equations 8 and 9 give the desired relationsl1iP as
9
1 1
I
'82
+ TZP + 1'3P
(10)
where
T z = (1
-htB/ 2QaC',)
(11)
Afhi
Ta = (1 + h!!!?k)w!!k!?! 2QaCa A / h f
(12)
There is also a transportation lag in the system which is equal to the time required for the air to flon- from the heater t o the thermal bulb ( 6 ) . At the normal flox rate 0 5 minute is required for the air to flow from the heater to the filter through tile plenum. Also, 0.22 minute is required for the air to flow through the 45 feet of exit duct, The transfer function for this dead time
The outlet air temperature is measured by a thermal --hi& produces an air pressure \J hich is directly proportional to the temperature of the outlet air. The selection and position of the thermal system in a revised transferfunction for each test. The thermal system used in Tests I and 11 may be represented as a single lag: (Test I )
AP 1 1 --- = __-~e~ 1 T,P
INDUSTRIAL AND ENGINEERING CHEMISTRY
+
(14)
Vol. 48, No. 6
PROCESS CONTROL the individual components, such as the heater or filter, have been sketched in lightly as described by the calculated transfer function. The over-all or combined calculated open loop response is shown by the heavy dotted line. The experimental response is shown by the heavy solid line passing through the plotted data. Test I was carried out with the temperature sensing bulb located immediately after the filter in the plenum as shown in Figure 1. As a result of low air velocity and insufficient mixing, control with the bulb a t this point was difficult. A very low sensitivity was required for stability, and the natural period of oscillation was long, thus requiring a long start-up period as
FREQUENCY RESPONSE DIAGRAM \
FREQUENCY RESPONSE DIAGRAM
CALCULATED-\
I
\
-300
I
I
\
Figure 3. Test I
(15) The value of Ta depends on t,he bulb construction and the air velocity past it. I n Test I the velocity was 20 feet/minute while in Test I1 the velocity was 200 feet/minute. T h e thermal system used in Test I11 used a different bulb which gave a larger time constant for the same air velocities. However, the transmitter introduced a derivative response. The transfer function for this unit is
-\
- 3 6 O L .001
1
The open loop response of the entire system, depending on the test, then becomes
The lags introduced by the pneumatic system are much smaller than those which have already been described; therefore, they were neglected.
Results and Discussion The calculated and experimental results for each test are given in Figures 3, 4, and 5 . T o show the effect of the position of the sensing element, the type of sensing element, and the transmitter used, we will examine each test. I n Figure 3, 4,and 5 straight line approximations have been used for the gainfrequency plots. The phase angle -frequency plots, however, have been constructed from calculated templates. I n each plot
June 1956
1
I
.01 01 1.0 FREQUENCY-CYCLES PER MIN.
Figure 4.
I
10
Test II
well as slow recovery following load disturbances. The experimental and calculated results are shown in Figure 3. Although the calculated result is not in exact agreement with the experimental result, the long natural period of oscillation and low gain experienced are easily predicted. If the operating point during the test of any nonlinear process element (such as the steam valve) differs from t h a t chosen for the calculation, the slope as taken from the characteristic curve will be in error. This is a possible explanation for the gain discrepancy noted in all tests. This does not limit the linearization method b u t indicates the importance of a more precise determination of the various operating points for each test. I n a n attempt t o reduce the natural period of oscillation of the system the bulb was moved t o a new location in the exit duct represented as Test I1 in Figure 1. At this point, 45 feet downstream from the plenum, both increased mixing and velocity were obtained. However, 45 feet of distance velocity lag was also introduced. Although a point closer t o the plenum would have
INDUSTRIAL AND ENGINEERING CHEMISTRY
1071
ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT made in the system have produced better control by ( 1 ) reducing the natural period of oscillation of the system and ( 2 ) by allowing increased controller gain, while maintaining the desired degree of stability. The close agreement between the analytical and experimental results indicates that the techniques used are reasonably correct. I n view of this agreement, it appears that this method could be usefully applied as a design tool to optimize the control results by integrating and analyzing all components of the control loop. I n order to strengthen this statement let us look a t our present system from the viewpoint of the design engineer.
FREQUENCY RES PONS E DI A G Fi AM
FREQUENCY RESPONSE DIAGRAM
I
0
z
,-60
Q W
u w K
u120 w
D
&-I80 cn Q
I
a
-240
-300+ -360‘ ,001
1 I .01 0.i IO FREQUENCY-CYCLES PER MIN
Figure
5.
IO
Test Ill
been desirable, the location used was the closest accessible point of installation. As a result of repositioning the temperature sensing bulb, the natural period of oscillation of the system was decreased from approximately 5.5 minutes to 4.0 minutes. Although the cutoff frequency of the bulb was improved from 0.074 ‘to 0.26 cycles/minute as a result of the increased air velocity, this improvement was partially off set by the undesirable addition s f the distance velocity lag. In an attempt to make additional improvements in the system a Transaire temperature transmitter with Speed-Act or derivative response (both Taylor instruments) was substituted in place of t h e original transmitter. With the substitution of the transmitter a neTT bulb v a s obtained as described in the theory. The location, the same as used in Test 11, is shcwn in Figure l as Test 111. As a result of the addition of the Speed-Act response in the transmitter the natural period of oscillation of the system was decreased from the 4.0 minutes of Test I1 to 2.9 minutes. Thus, by merely repositioning the sensing element, as in Test 11, and making a change in the transmitter, as in Test I11 the natural period of oscillation of the system was decreased from the original value of 5.5 t o 2.9 minutes. At this point let us review what has actually been accomplished. From Test I and I1 we have shown the effect of sensing element position, the advantage of the increased velocity across the bulb being partialh. offset by the disadvantage of the added distance velocity lag. Test I11 s h o w the importance of selecting the proper transmitter and also, by comparison with Test 11, points out the difference in the two transmitter bulbs Actually, no revolutionary discovery has been made, nor has anything been done analvtically or experimentally that probably could not have been predicted by an eyperienced control engineer. The changes 1072
\
- 3 6 0 1 L A , , ,001
Ql 1.0 FREQUENCY-CYCLES PER MIN.
.01
IO
Figure 6. Calculated results for a possible process design change
I n any design xork, compromises must be made between such variables as space, weight, materials, and ideal system. This is true regardless of the problem, and the designer w-ho has the informahion before him must make the final choice. However, as a n example of a possible design change t h a t would result in improved control of the present system, let us examine the filter. If it had been possible to reposition the filter outside the control loop, two things would have been accomplished. T o begin with, the filter transfer function mould be entirely removed from our analysis. The plenum chamber section would have been replaced by a length of smaller exit duct work resulting in increased air velocity. XOK let LIS examine Test I in the light of this change. Our first change would be a reduction in distancevelocity lag from 0.5 minute (10-foot length of plenum chamber having a velocity of 20 feetjminute) t’o 0.05 minute (10-foot length of exit duct with a velocity of 200 feet/minute). A t the same time, the increased velocity would result in giving the bulb response shown in Test 11. Calculated results for this supposed system change are shown and compared with calculated results of Test I in Figure 6. According t o these calculations the design
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 48, No. 6
PROCESS CONTROL revision has changed the natural period of oscillation of the system from approximately 5.5 to 0.5 minutes. At the same time allowable controller gain on the basis of a 30" phase margin shows an increase from 2.5 t o 3.5. These figures speak for themselves and prove t h e value of this tool t o the process designer. I n making this type of a n analysis, however, one must remember t h a t we have linearized our system by assuming small changes around a n operating point. Thus, in order t o avoid misleading conclusions, the designer must check his linearized calculations at a number of points within the limits of his expected operating range.
h = film coefficient
( h h = 5.5, hj = 2, B.t.u./hr.-sq. ft.-" m = latent heat of steam (932 B.t.u./lb.) L = dead time of transportation lag, min.
p
d dt
= heaviside operator = - (l/min.)
Literature Cited
Conclusion
A method has been presented for analyzing processes and the associated control components. Improvements on a n existing process were predicted and verified experimentally. A possible reorientation of a process variable to provide a major control improvement is predicted. Although this prediction has not been verified experimentally, for obvious reasons, the value of dynamic analysis as a tool to the process designer has been shown. Ackno,wledgment
The authors appreciate the assistance and cooperation shown t o them by the personnel and management of the Bell and Howell Co. of Rochester, N. Y . ,without which this study would not have been possible. T h e assistance of Mary Jeffries and Raymond Johnson in the preparation of the manuscript and drawings is also noted. Nomenclature a
P Y A
E
A
c
=
aQ* (157 lb./hr.-lb./sq. dP,
inch)
!& (-19 Ib./hr.-lb./sq. inch) = ? ap. (0.45 lb./sq. inch/" F.) = 3 hi3= s z k l change ;f a variable = temperature, F. = outside areas (Ah = 1170 sq. ft., A , = 2800 sq. f t . ) = specific heat ( c h = 0.12, C f = 0.1, c, = 0.24 B.t.u./lb.F.) = sensitivity of heater (12.9" F./lb./sy. inch)
F.)
Aikman, A. R., Trans. Am. Sac. Xech. Engrs. 76, 1313-23 (1954). Aikman, A. R., Trans. Sac. Instrument Techml. 3, 2-16 (1951). Brown, G. S., Campbell, D. P., "Principles of Servomechanisms," p. 143, Wiley, New York, 1948. Deisler, P. F., Jr., Wilhelm, R. H., IND. ENG.CHEM.45, 1219-27 (1963). Eckman, D. P., J. Instrument Sac. Amer. 1, No. 12, 11-14 (1954).
Farrington, G. H., "Fundamentals of Automatic Control," p . 152, Chapman, London, 1951. Fiedler, G. J., McGrath, T. F., Buescher, A. E., Control Enaineering 2, No. 7, 65-80 (1955). Holzmann, E. G., ASME Paper No. 55-IRD-5, Instruments and Regulators Division Conference, Ann Arbor, kfich., April 1955. Hoyt, P. R., Stanton, B. D., Imtrunzents 26, 1180 (1953). Hoyt, P. R., Stanton, B. D., Symposium on Instrumentation for Process Industries, College Station, Tex., January 1953. Long, RII. V., Holzmann, E. G., Trans. Am. Soc. iMech. Engr. 75, 1373-81 (1953). Macmillian, R. H., Trans. A m . SOC.Mech. Engrs. 76, 1237-44 (1954). More, J. L., Quail, F. J., Bain, J. W., Isn. ENG.Cximi. 37, 91216 (1945). Paynter, H. M., Takahashi, Y., ASLIE Paper S o . 55-SA-50, ASME Semiannual Meeting, Boston, Mass., June 1955. St. Clair, D. W., Erath, L. W., Gillespie, S. L., Trans. Am. SOC. Mech. Engrs. 76, 1177-84 (1954). Takahashi, Yasundo, Control Engineering 2, 4G50 (1955). Taylor Technology 4, 14-17 (Spring 1952). Young, A. J., Instruments 26, 878 (1953). Ziegler, J. G., Nichols, N. B., Trans. Am. Soc. Mech. Engrs. 65, 433-44 (1943).
O
B
RECEIVISD for review Janunry 21, 1956.
-4CCEPTED March
28, 1956.
Monsanto Chemical Co. computer installation has 8 tape units, 2400 feet of magnetic tape, which handle information equivalent to that storable on 25,000 punched cards June 1L956
INDUSTRIAL AND ENGINEERING CHEMISTRY
1013
