COURTESY.
An outline i s presented of t h e methods used by Ziegler and Nichols for finding t h e optimum settings for temperature controllers. Applied t o a heat exchanger, coefficient u t o be used in t h e sensitivity equation, S = u/&L, was determined by t h e reaction or response curve method. T h e work was carried further by calculating t h e value of the capacitances of t h e exchanger which contribute t o the form of t h e response curve and by graphically reconstructing this curve. Coefficient K found by this method compares favorably with t h e value determined experimentally.
rnE LUMMUS
COMPANY
therefore, be described only briefly in this paper. Generally, the most desirable controller setting for a given process is that a t which the recovery curve obtained after any disturbance of the equilibrium of the process shows an amplitude ratio (ratio of magnitude of each oscillation to that of the oscillation preceding it) of 25%. Therefore, the three methods described below are intended to indicate the proper controller setting to give this 25% amplitude ratio. ULTIMATE SENSITIVITY AND RESPONSE CURVE
I n most processes a controller sensitivity (output change per unit of pen movement) of one half that at which an amplitude ratio of 100% is obtained (ultimate sensitivity) gives a recovery curve having an amplitude ratio of approximately 25%. Therefore, to determine the optimum setting by this method, it is necessary only to find a sensitivity at which sustained oscillation results from a process disturbance and set the instrument a t half of that sensitivity. This method is applicable only to multiplecapacity processes, but as most industrial processes come under this classification, no great disadvantage is presented. The only drawback t o the use of the ultimate sensitivity method lies in the time required to make the necessary tests. I n large-scale processeshaving large capacities, the period of oscillation becomes very long, and determinations are time consuming. The response curve for a process is obtained by suddenly changing the flow of the control agent and plotting the resulting changes in the controlled variable on a time base, in the absence of any controlling action. I n practically all industrial processes the response curve is S-shaped (Figure 1). A tangent drawn through the point of inflection gives an indication of the characteristics of the process and serves as a am for determining optimum settings for its control.
HE refinements introduced into the design of industrial automatic control instruments in recent years, and the consequent rationalization of the knowledge of their characteristics, have resulted in a more systematic study of practical applications. The object of this study was to devise means of determining optimum controller settings for the various conditions of lag and capacitance which make up an industrial process. The work of Ziegler and Nichols ( 4 ) resulted in the development of three means of determining optimum controller settings: ultimate sensitivity, response curve, and analytical response curve methods. The work described here was performed (a; to determine constants applicable to the response curve method when used to determine optimum settings for industrial heat exchangers, and (b) to evaluate effects of the various design factors of heat exchangers on the controllability of the process and thus make possible the application of the analytical response curve method to problems of industrial heat exchanger control. The ultimate sensitivity and response curve methods for determining automatic controller settings and their application to a distillstion process were reported by Allen ( I ) . They will, 912
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F.J. QUAIL A N D J. W. B A I N '
L. MORE
COMPANY. DEAUPRP,
auesasc, CWAPA
U N I V S ( P B l t Y OF TORONTq. CANADA
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When B unit change in the control agent is madc--c.p., lira1 caused by B change of I pound per square inch in air pressure on a diaphragm valve-tho product of the slopc of the tangent to the p i n t of inRection oi the rosponse curve, R,, and ita intercept on the horizont-al axis, L, gives a direct reciprocal relation for the determinstion of o p t i m a sensitivity of a controller applied t o the process. L, the lag, is approximately constant for the proce s , regardless of the change made in the flow of the cont,rol agent; the reaction rate varies directly RS Ihe chaiige rnade in the Row of the control agent and must ir eoiwerl.cd t o a. unit reaction rate in optimum setting determinations. The product, R,L, far B process is repreasanted by the intercept of the tangent, through the point of inflection with the vertical axis. Although the response curve method eivca B direct reciprocal relation for determining optimum controller settings, many induvtrial proresses require B iactor to be applied to tho R L values. The general formula. for obtaining optimum settings iron, response curve is:
s = urtz,i.
where u is z. con~tantwhich must be found experimentally for the various types of procesula. Values for u have been reported an high as 5.4 on pure two-capacity proeexses, although most iirdustrihl processes show B value newer unity. The response curve method luls one disadvantage in practice: I t frequently requires a relatively lame disturbance in tho process, and such disturbances cannot be made in many processes for praetiosl and economic r e ~ ~ o n s ANALYTICAL RESPONSE CURVE
method involves calculntion MATHPIMATICAL A ~ n ~ r s l sThis . the response curve irom calculated values of capacitance, &stance, and t r ~ n s p o r tlag which make up the proeerii. This determination may be either by purely analytical means or by of
gctphical methods. Calculation of analytical response curves may be illuatratad by considering a pure two-espscity process. Figure 2 shows B pro~essin which the water in tank I is heated directly by a stam jut. and that in tank I1 is heated by the overflow from tank I. k u m i n g perieet mixing in both tanks arid neglecting the small resistance of the tliormorneter bulb and tho transport lag be-
I N D U S TRI A L A N D E N G I N E ERIN G CHEMISTRY
914
instant and the ultimate temperature, makes it possible to calculate this curve. Let t = T = TO = K1 = Ka =
temperature a t any instant in tank I ultimate temperature initial or datum temperature time constant for tank I min. time constant for tank fI, min. = time
e
bnyndx
- nxyn-idy - naynay = 0
Let bn = -1,
(T - t ) / K i
T-t
+ aKldy (fqom Equation 2)
then
Let Ka/Ki = b and multiply by ny"-l; then,
Then for tank I, Newton's law may be expressed as followa: dlld8
Vol. 37, No. 10
:. n = -I/b
i
and
-a(ynx)
na -dy"+' n + l
=0
Integrating,
:.
2
= Cay-"
na -n+1Y
(3)
but
1
n=-T=-Figure 1.
Ki Kz
Unit Response Curve
Integrating, -In (T
n :. n+ l
- t) = 8/K1 + C1
when 8 = 0, O/K1 = 0, and t = TO. Therefore Cl = - h ( T TO)and --In (2' 1 ) = ~ / K I ln(T TO), which becomes
-
-
-
-
Ki -Ka
- RI
Substituting in Equation 3,
and
t
or
- TO= a;
Let T
=
T
- ( T - To)e-e/Ki
C, can be evaluated from the fact that, when
- ae-e/Ki
(1)
which is the equation for the temperature of tank I a t any instant. Considering tank 11, -dt'= - t d8
where t t'
K1
where 1' Then,
K,
constant for tank I1
.. K&' but
- t'
= temperature of tank I = temperature of tank I1 =
p
td8
- t'd8
- ae-@/Ki (from Equation 1) .'. K&' = T d 8 - ae-@/Kd8 - t'd8 Kldt' = ( T - t')dO - aeWK1dO and 8 are variable. Let t' - T = x, and e-e/K1 t
=
T
at' = ax e-@/Kid8 = -Kldy
and
de = -K1-dY Y
8 = 0, t' = TO.
R and L can be calculated from Equation 4, but the method is
then
t =T
(4)
(2) =
y.
cumbersome. The addition of a third capacity further complicates the final equation, and its usefulness disappears. GRAPBICAL ANALYSIS. The substitution of values in complex equations can be avoided by graphically constructing response curves of processes whose time constants are known. Such curves (Figure 3) are close enough approximations to tha true response curves for optimum setting determinations and are much more convenient than the purely mathematical analysis. Figure 3 was drawn for a three-capacity process with time constants of 4, 4, and 8 minutes. Curve I represents the response of the first capacitance (& = 8 minutes). If, a t any time during a change in temperature, the temperature is allowed to rise a t a constant rate rather than a t a rate proportional to the temperature difference, the temperature curve becomes linear. Curve I was constructed by imagining a series of constant-rate ttunperature rises over small time intervals; i.e., A to B may be prolonged to C where the time between C and C' (where C' is the average time during the change A to B ) is 8 minutes (= &). Time increment A to B is 1.0 minute. The next addition to the curve starts at B and is aimed towprds E. This addition is represented by BC,the average time is represented by point E', and the time from E' to E is again equal to K1, the time constant for the first capacity. Each of these short lines approximates the corresponding portion of the exponential ourve which would represent more truly the response of a capacitance having a time constant of 8 minutes.
oatober, 1945
INDUSTRIAL A N D ENQINEERINQ CHEMISTRY
-
Curve I1 r e p m n t s the response of the second capacitance 4 minutes) and is simiIar to curve I except that each of the short straight lines is directed toward the point on curve I with an abscisea 4 minutes removed from the mid-point of the line. This is because the potential temperature of fhe second capacitance st any instant is the temperature of the first capacitance at that same instant; e.g., F is 4 minutes removed from A, FG is drawn toward point C', the mid-point between F and G being 4 minutes from C', etc. Once again each time increment is 1.0 minute. Curve I11 represents the response of the third capacitance (Ka= 4 minutes) and is similar to the other two, except that the linea are directed towards corresponding points on curve 11. This represents the response curve which would be obtained from a p r o m having the characteristics dwcribed; from it the value of RIL can be found in the manner described for experimental reeponse curves. When transport lags are p m n t , they are included by adding a horizontal portion to the curve along the line of TO. The final curve is dmwn only as far as the point of inflection, which is easily found by noting the changes in inclination of the straightedge used to construct it. In the application of this method, the time constants can be used in any order without changing the final result, but it is generally more convenient to plot the curves in the order of descending magnitude of time constant. Either of the modifications of the analytical response curve method for determining optimum settings can be used without experimental work, if constant u and the governing capacitances
(Kr
915
of the particular type of process are known. The following work is concerned with the evaluation of these factors for tubular heat exohangem. DESCRIPTION
O F HEAT EXCHANGER
The heat exchanger used for this investigation is shown in Figure 4 and in the photographs on page 915. The outer chest is made of cast,steel; i t is 6 inches in diameter and has an over-all length of about 6 feet. Cold water enters from the left into the left-hand reservoir of the exchanger, and p a m a through four inner 1-inch steel tubes, where it is heated by the surrounding steam, to the right-hand reservoir or mixing chamber and then out through the outlet pipe past the temperature controller bulb. The steam is pawd 5rst through a dryer and then through the circuit containing the control valve to the steam chest where it heats the water in the inner tubes. The resulting condensate collects in the chamber a t the bottom (Figure 4). Constant water flow is attained by a large constant-head tank (not shown in Figure 4). The rate of water flow was measured by a calibrated rotameter. The upstream steam pressure was adjusted manually and waa held constant throughout all the experimental work. The control valve was of the air-to-close beveled disk type with a diaphragm motor. A Precisor was attached to ensure adequate response to changes in air pressure. Temperature was recorded and controlled by E Taylor Fulscope recording controller, equipped .with proportional response. DETERMINATION OF SENSITIVITY
STbAM
-=-=lil
I
TRIAL-AND-ERROB MEITHOD.The actual sensitivity setting required for satisfactory control wtq determined by making a series of settings of the sensitivity dial; for each value of the sensitivity selected, the system was upset by causing a sudden change in the heat supply. To do this, the control point was changed abruptly and then readjusted to ita former value. The result of this operation gave the familiar wavelike tracing of temperature change about the control point. Sensitivity S, TANK f TANK which gave a die-away curve having a 25% amplitude ratio, was to be 7.8 lb. per sq. in./in. This value waa taken as the found Example of Simple Two-Capaoity Pmcert mwt desirable sensitivity setting for the Drocess. - RrcsA~s~ CURVEMETHOD. Under conditions of constant outlet water temperature, and with the controller acting as a temperature recorder only and not as a controller, the air pressure to the diaphragm " ' K,*4m. of the control valve was suddenly changed. This resulted in E corresponding alteration in the sterwn supply and, hence, a change in the temperature of the outgoing water. To facilitab the plotting of the temperature response curve on rectilinear coordinates, the clock in the recorder waa stopped, and the temperature chart of the instrument rotated by hand every 6 seconh. This resulted in a stepwise tracing of temperature v8. 6 - s e c o n d intervals. A plot of the data obtained in this way gave the characteristic S-shaped curve. Figure 3. Graphical Construction of a U n i t Response Curve
II
Figure 2.
a
916
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 37, No. 10
Recorder Bulb and Mechanism. During the early stages of the experimental work i t was found necessary to reduce the sensitiveness of the bulb to small temperature fluctuations by wrapping the bare bulb with friction tape. Under normal conditions the time lag of the recorder and bulb is about 2 or 3 seconds. The time lag of the wrapped bulb was found by immersing i t in hot water and then quickly placing i t in position at the outlet of the exchanger, with cold water running through the tubes. The timetemperature curve was plotted on rectilinear coordinates, and the bulb constant determined by noting the time corresponding to 63.2% of the temperature change (8). This value was found t o be 20 seconds. Distance-Velocity or Transport Lag. The transport lag is made up of the time for the water to flow through the right-hand reservoir to the pipe leading up to the recorder bulb and the time for the water to flow up this pipe to the bulb. The total transport lag was found to be 10 seconds. The three capacity lags were 45, 25, and 20 seconds, and the transport lag was 10 seconds. A response curve was constructed e8 indicated by Figure 3, and sensitivity S was found to be 8.1 lb. per sq. in./in, If the trialand-error value of 7.8 lb. per sq. in./in. is assumed to be the true sensitivity setting, then u in the sensitivity equation becomes 0.96, which compares favorably with 0.91 obtained by the response curve method.
TEMPEMNRE C0NTfiOL.f E8 R€COBO€R
-
J7EAM M I C R
CONOENSATE CCiU-CmR
Figure 4.
Diagram of Heat Exchange Setup
RiL was found to be 5.3’ F./lb. per sq. in., and hence the sensitivity is: S = 45.6’/5.3 = 8.6 lb. per sq. in./in. where 45.6” F./in. converts from degrees Fahrenheit to inches for this particular type of chart. Comparing this value with S found by the trial-and-error method gives 0.91 for u, and the sensitivity equation becomes S = O.Ql/RLL Ib. per sq. in./in. ANALYTICAL METHOD. Tubular heat exchangers consist of a complex system of lags. Besides the heat capacities of the various elements, there is the transfer lag governed by the heat conductivity of the tubes, and the transport lag due to locations of control valves and thermometer bulbs (a). T o represent such a process by a small number of capacitances and transport lags involves lumping some capacitances and neglecting others which have little effect on controllability. In the present case the system was considered to contain four major lags: the exchanger jacket and tubes, the water in the tubes, the recorder bulb and mechanism, and the transport lag due to the time for water to flow from tubes to thermometer bulb. Of these four lags, only the third was found by experiment. Time Lag of Steam Jacket and Tubes. The total combined weight of steam chest and tubes was 120 pounds. The specific heat was 0.12 B.t.u./lb./’ F. The heat capacity of tubes and chest was 120 X 0.12 = 14.5 B.t.u./O F. If i t is assumed that the exchanger has been operating under constant conditions, and the heat supply is changed by y B.t.u./sec., then the jacket and tubes will rise at a rate of y/14.5’ F./sec. Water is flowing through the tubes a t a rate of 0.305 lb./sec. and will rise a total of y/0.305” F. to the new temperature level. The capacity lag (or time lag) of the jacket and tubes may now be calculated as (y/0.305)/(y/14.5) = 48.3 seconds (say, 45 seconds). Time Lag of Water in Tubes. The volume of water in the tubes a t any instant is 0.12 cubic foot. The weight of water at any instant is 0.12 X 62 = 7.5pounds. The water will rise a t a rate of y/7.5’ F./sec. to the new potential, which is y/0.305’ F. From this, the time constant is (y/0.305)/(1//7.5) = 24.6 seconds (say, 25 seconds).
CONCLUSION
The summary of results by the three methods follows: Method Trial and error Res omacurve An8?yhiO81
5. Lb. per Bq. In./In. 7.8
8.6 8.1
21
0:bl 0.96
The term “process sensitivity” ( 8 ) was suggested by J. G. Ziegler. Its units are inches per pound per square. inch (in./lb. per sq. in.) and refer to the inches of pen travel per pound per square inch change in air pressure. The vertical axis of Figure 3 employs this unit, which may be made as large as is desirable. RIL is then multiplied by the ratio of the actual process sensitivity to the scale sensitivity. The actual process sensitivity may be calculated and is constant for any specific process. Only the flow-lift characteristics of the control valve must be known. For this work an experimental value of 0.402 in./lb. per sq. in. was used, and a scale value of 6 in./lb. per sq. in. ACKNOWLEDGMENT
The authors wish to thank K. B. Jackson of the Department of Applied Physics, University of Toronto, and J. G. Ziegler, of Taylor Instrument Companies, for their assistance; also the Taylor Instrument Companies for the use of their equipment. LITERATURE CITED
(1) Allen, L. H., Jr., IND.ENQ.CHEM.,35, 1223 (1943). (2) Beck, R., Trans. Am. SOC.Me&. Engrs., 63,531-33 (1941). (3) Rhodes, T. J., “Industrial Instruments for Measurement and Control”, Chap. I X , New York, McGraw-Hill Book Co., 1941. (4) Ziegler. J. G., and Nichols, N. B., Trans. Am. Soe. Mech. Enom.. 64,759-68 (1912); 65,433-44 (1943).
PARTof a thesis submitted by J. L. More and B.J. Quai1in partial fulfdment of the requirements for the degree of B.A.Sc. in Chemioal Engineering, University of Toronto, 1944.
