ANALOGS which near optimum instrument settings could be empirically determined. Although the final control valve had an appreciable hysteresis zone, the process could be satisfactorily controlled by reducing the proportional band.
q2
R r S T 1 94.1
Nomenclature
= outflow from upper tank, cu. ft./sec. = flow resistance of weir, sec./sq. ft. = reset rate, min.-1
= proportional band,
%
= time constant, see.
= time, see. = load variable, cu. ft./sec.
Subscripts
A
=
As
= percentage change in measured variable that produces
area of tank, sq. ft.
reaction curve C = capacity, sq. f t . c = controlled variable a t time, t, ft. cf = final value of controlled variable, ft. g = derivative time, min. h = head of water a t time, f, ft. h, = head a t initial conditions, ft. h, = head a t final conditions, ft. K, = gain, ratio of output to input L = lag from reaction curve, see. A M = percentage change in valve position required to produce reaction curve m = manipulated variable, cu. ft./sec. N = reaction rate, % scale/sec. po = outflow from lower tank, cu. ft./sec.
1 = lower tank 2 = upper tank
o = initial conditions = final conditions
f
Literature Cited (1) Chien, K. L., Hrones, J. il., and Reswick, J. B., Trans. Am. SCC. Mech. Engrs., 74, 175-85 (1952). (2) Cohen, G. H., and Coon, G. A., Ibid., 75, 827-34 (1953).
(3) Pratt, E. A., Eng. News, 72, 462-3 (Aug. 27, 1914). (4) Ziegler, J. G., and Nichols, N. B., Trans. Am. SCC. Mech. Engrs., 64,8,759 (1942). RECEIVEDfor review September 16, 1954. ACCEPTED J a n u a r y 18, 1955. Abstracted in part from a thesis submitted by P. J. Sauer to Case Institute of Technology in partial fulfillment of the requirements for the Master of Science degree in Chemical Engineering.
Applications of Pneumatic Analog ERNEST F. JOHNSON
AND THEODOSIOS
Princeton Universify, Princefon,
BAY
N. 1.
I n studying process control theory it i s advantageous to use a simple analog which behaves like commercial processes. The pneumatic analog used in this work consists of an array of volume chambers connected b y fine bore tubing dead-ended at a conventional pneumatic recording controller, which feeds air to the volume chambers through a null-balance pressure regulator. Experimental frequency response characteristics of a three-stage process are compared with characteristics predicted from the system geometry and flow equations. Three different methods for predicting desirable control settings for proportional and proportional plus reset control are applied to the process using both frequency and transient response data. The particular advantages and the practical limitations of pneumatic analogs are discussed and compared with other types such as hydraulic and electrical analogs.
C
HEMICAL engineers are beginning to realize the importance
of a quantitative understanding of the principles of automatic process control. Certainly the successful performance of particular processes and component apparatus depends as much on the effectiveness of regulation as it does on the proper design of major units involved in the process. As our technology increases in complexity and processes are speeded up, the control aspect grows in relative importance, and as a result the empirical practices used in instrumenting processes in the past become inadequate. Because empirical instrumentation has been successful in the past, the chemical engineer has not concerned himself with control theory, and the main theoretical advances have been made by electrical and mechanical engineers. Unfortunately most of the developments in control theory have been applied to problems of little interest to the chemical engineer, and, although the principals involved are of general applicability, the language used is unintelligible to him. A major problem is to translate the pertinent theory of communications and servomechanisms into the working language of chemical engineering. Some attempts have been made to effect this translation (1,f?,4, Q), but much remains to be done. This paper describes t h e results of a study undertaken t o explore a t a simple level some of the pertinent concepts
March-1955
of control theory as applied t o a process familiar to chemical engineers. To avoid the need for large scale operation of a prototype process, a small pneumatic analog involving the flow of air through a series of tanks and capillary tubing was used in conjunction with a conventional pneumatic controller and pressure regulator. The pneumatic analog was chosen over an electrical analog to permit the use of a more familiar medium and over a hydraulic analog to permit simpler and neater manipulation. The dynamic characteristics of the controller and analog were determined experimentally and also computed theoretically. Despite the use of simplifying assumptions the general agreement between predicted and actual behavior is good. I n a simple exposition of the use of these data, three different methods for predicting controller settings are examined and compared. Theoretical All the important operations of chemical engineering may be described in a simple manner in terms of
1. A flow of matter or energy which is proportional to a. The drop in driving force or potential for t h a t flow and b.
A reciprocal resistance to flow
INDUSTRIAL AND ENGINEERING CHEMISTRY
403
ENGINEERING, DESION, AND PROCESS DEVELOPMENT 2. A storage of energy or matter which depends on the capacitance of the system
Thus simple systems may be represented as various serial combinations of reeistances and capacitances through which energy or matter flows. Each combination of resistance and capacitance is called a n RC stage or a first-order transfer stage, with behavior given by (4,6)
where s 01 62
djdt, a n operator and root of the corresponding characteristic equation = potential before resistance = potential in capacitance =
The theoretical transient characteriqtics of the procejs alone may be obtained from Equation 3 by setting 81 = A , the input step, and solving the resulting differential equation. Experimental transient response curves for process are sometimes called reaction curves or signature curves. They are obtained by Applying a sustained step change to the proce;is input and plotting the output change with time. The combined transient behavior for the process analog arid controller niay be obtained theoretically by relating process input and output, 61 and e2, through the equation for the controller, inserting in Equation 3, and solving for a step change in controller set point. For a three-mode pneumatic controller of the type used in this work, with the rate or derivative action set a t zero ( 8 )
Since the product RC has the dimensions of time, it is frequently called the time constant, T , of the system. where
e1
= controller input or difference between controller set
81 =
-
1
RI
MOORE VALVE
Figure 1 .
i
Cl
Schematic diagram ,of three-stage pneumatic analog
Equation 1 niay be written
and for three similar stages coupled together in such a manner that the downstream stages cannot affect the upstream stages
ea
Resistances may be computed from Poiseuille's law for flow through capillaries assuming an average air density, and capacitances may be obtained from the volume of the chambers and the average air density. Thus,
R = 404
--
*gpav.d4
(5)
and
The transient response obtained above would represent the closed loop behavior of process plus controller. A more flexible characterization of dynamic process behavior is through frequency response analysis. I n general if a steadystate sinusoidal signal is impressed on some input to a process, the corresponding output signal v d l vary sinusoidally a t the same frequency as the input wave hut with a different, usually smaller, amplitude and displaced in phase, usually lagging. The frequency response characteristics are the ratio of output t o input aniplit,udes (magnitude ratio or gain) and t'he phase angle bph e e n them over ranges of frequency. These charact,eristics are most effectively used in terms of over-all open loop behavior, which may be obt'ained merely by adding logarithms of amplitudc ratio and numerical values of phase angle of the individual coniponents of the loop. This simple additivity of component characteristics is one of the principle advantages of frequency response techniques. Since practical design rules have been worked out, in terms of open loop characteristirs, it becomes a relatively simple matter to gage the effect of various types of controls on a given process, provided the frequency response dnta are available or readilv determinable. Theoretical frequency response characteristics n a y be obtained by setting s = j u , in t,he differential equation deeoribing the and w = the angular frequency, radians system where j = per second, of the input sine wave. By separating the resulting equation into its real and imaginary part,s, the ratio of output and input amplitudes and the phase angle between them may be found as functions of frequency. For a single first-order transfer stage (Equat,ion 2 ) the amplitude ratio or magnitude ratio is given by
47
1
This equation also describes the case vdiere the stages interact with each other provided t h a t cffective time constants are chosen in terms of appropriate combinations of RC products ( 4 ) . For a pneumatic analog involving volume chambers separated by capillary tubing restrictors the transfer stages are interacting and effective time constants must be used in Equation 3. I n terms of the individual time constants, 1edstances, and capacitances for a three-stage analog Equation 3 becomes after expanding the denominator
128~1
point and measured variable e4, Ib./sq. inch controller output, pressure, 1b.jsq. inch
C
=
VAP
(6)
and the phase angle, q ,is equal t o arctan w T 1 . Experimental frequency response data are obtained by applying a steadystate sinusoidal variation to thc process input and comparing inlet and outlet waves as to amplitude and phase angle. At low frequencies it is often simpler to apply a rectangular Ivave input signal. If the process involves one or more first-order stages, the input TTave may be assumed to be a pure sine xave equal t o that given by the fundamental frequency term in the Fourier series reprcsenting a rectangular wave. Apparatus
The analog apparatus used in this study was based on a simplification of an earlier design (6). It consists of various capped lengths of standard 2- and 1.5-inch pipe connected by
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 47, No. 3
ANALOGS means of a/,inch copper tubing and various lengths of 1-mm. capillary tubing. Figure 1 shows the control loop schematically. The largest capacity, CI, is made up of lengths of 2- and 1.5-inch pipe having a total volume of 0.2795 cu. ft. Capacity, Cz, is a single length of 1.5-inch pipe with a 0,0258-cu. ft. volume, and the smallest capacity is the pressure measuring bellows in the controller with connecting tubing having a combined volume of 0.00271 cu. ft. The resistances before and after the middle capacity, RS and R3, are 6-ft. lengths of capillary tubing, and E1 is a 3-ft. length.
Table 1.
Controller Calibration
Proportional Band Settings, % Actual 0-20 inch Lb./sq.
Theoretical 10 20 30 40 50 60 70 100
Lb./sq. 3-15,inch
14.1 29.2 46.5 61.8
8.5 17.4 28.4 37.1 46.2 57.3 64.7 93.2
77.1 96.6
...
... Reset Rate, Repeats/Min. Actual 0-20 Lb./sq. inch
R
Theoretical
3-15
Lb./sq. inch 0.120 0.151 0.252 0.344 0.445 0.657 1.065 1.62 2.56
0.118 0.148 0.248 0.338 0.439 0.645 1.046 1.59 2.52
120
-
130
UJ W
W a
J 140 n
w J
IS0
(3
2
a
a Figure 2. Process frequency response characteristics-magnitude ratio
X
0
Experimental, re-tangular wave Experimental, sine wave
- Theoretical, sine wave --Theoretical, sine wave approximation
\ \
-I
\
2 170 W
\ \
I
a
\
\
X
\
180.
\
\
\ \
*
The controller is a Bristol Series 500 three-mode pneumatic pressure recorder and controller. This instrument has two recording pens, a circular chart driven a t 2 revolutions per hour, and a range of 0 t o 20 pounds per square inch. A Moore nullbalance pressure regulator serves as the final control element. By loading its diaphragm with the output pressure from the controller instead of a spring, a one to one correspondence between controller output and process input is obtained without attenuation and only small phase shift over the ranges of frequencies which are important to the process. A small volume chamber is installed between the controller and the regulator to prevent high frequency pneumatic resonance. For frequency response studies a simple sine wave generator is connected to the set point index of the controller by means of a gear and rack. The generator consists merely of a cam with adjustable eccentrics, coupled to the rack and driven by a small variable-speed motor and a reduction gear train.
Experimental The pneumatic controller was calibrated by noting the effect on output of changing the set point a t constant controller inputs for various settings of proportional band and reset rate. Table I summarizes the results. The calibration results are given for effeotive output pressure ranges of 0 to 20 pounds per square inch gage used by the Moore
March 1955
sol
I
.04
I
I
/
.OS .08 .I FREQUENCY
I
2
I
.3
1
A
(RADIANS ISEC.)
Figure 3. Process frequency response characteristics-phase angle X
Experimental, rectangular wave
0 Experimental, sine wave
- Theoretical, sine wove _--Experimental, corrected for components between set point and process input
regulator on the analog and 3 to 15 pounds per square inch gage used by commercial control valves. Since the controller was designed to be used with commercial control valves, the calibration based on the 3 to 15 pounds per square inch gage range agrees more closely with the instrument scale markings. The frequency response characteristics of the process were obtained from manually applied rectangular wave input signals and mechanically applied sinusoidal input signals. The data are listed in Table I1 and plotted in the form of Bode diagrams in Figures 2 and 3 together with the corresponding theoretical amplitude ratios and phase angles computed from Equations 4,5, and 6. For comparison the straight line approximations to the theoretical amplitude ratios based on the corner frequencies a t reciprocal effective time constants ( 3 ) are shown as dashed lines in the figures. The basis for these approximations can be seen
INDUSTRIAL AND ENGINEERING CHEMISTRY
405
ENGINEERING, DESIGN, A N D PROCESS DEVELOPMENT
Table 11.
Experimental Frequency Response Characteristics of Analog Process
Frequency, Radians/ Sec.
0.0627 0 07 0.0836
0 1048 0.1256 0.157 0 1673 0 2099
Amplitude Ratio 0.121 0 0982 0.0766 0 0755 0.0794 0 0681 0,0530 0.0504 0.0504 0.0393 0 0354 0 0378
Input .4mplitude, Lb./Sq. I n c h 2 2 2 2 2 3 2 3 3 2 3 3 2 3 5 3 3
Phase Angle, - 122 - 125 -127 - 130 - 132 - 128 - 143 - 136 - 137 - 144 - 149 - 145 - 146
0,0227 0.0199 0.0181 0.0247 0 0151
-100 - 180
- 150 - 176
Between Set Point and Process 1.6 1 6 1.6 1.6 1.6
by writing Equation 3 as the magnitude ratio for steady-ctate cycling,
If TO< Tg < TI then for 0 < w < 1 / T the magnitlide ratio approximates unity; for l i T < w < 1/T2it falls off a t a slope of -1 on a log-log plot of magnitude ratio versus frequency; for 1/Tz < < 1/T3 it falls off a t a slope of -2; and for w > 1/Ta it falls off a t a slope of -3. At very loiv and very high frequencies the approximation holds well, and for intermediate fucquencies the actual behavior can be fixed from a few computed points. The dashed line on the phase angle curve for the process, Figure 3 , includes the additional lag due to the system components between the controller set point and the process input These components had no measurable effect on the gain. The system resistances. capacitances, and time constants used in computing the theoretical frequency response characteristics are listed in Table I11
Table 111. Stage number
System Parameters 2
3
101,700
101,700
1.3171 3 . 4lo-'
1.41
13.4
1.26
1
Resistance, R , lb./sq. inch sec./lb. 5 0 , 830 Capacitance, C , Ib./lh./sg. inch Stage timeponatant, T ,sec. 1 4 , 2752 . 4lo-' Effective time constant. T. sec. 81.0
minute. If the theoretical ultimate proportional band and frequency are computed allowing for the observed phase angle between controller set point input and process input, the results are 3.7% and 1.4 cycles per minute. Discussion The agreement, betn-een experimenta,l and theoretical results is good, particularly in view of the various possible errors. I t was estimated that the experimental values of amplitude ratio could be in error due to t,he manner of recording and extracting much as + l o % a t low frequencies to, data from the charts b? +377, at, high frequencies. Similar estimates for phase angles n-ere i.7 and &16%, respect,ively. Equally great potential errors were estimated for the theoretical frequency response characterietics due primarily to uncertainties in the diameter of the capillary tubing. The aesumptione of lumped resistances and capacitances and linear behavior are believed t o eont,rihute negligible error because of the design of the analog and the low amplitudes used. Most of the frequency response characteristics were determined from rectangular rat'her than sinusoidal 'rvave inputs, and, although a preliminary theoretical st,udy had shown that the rectangular wave behaves for all practical purposes like a sine wave having an amplitude 4 / T times the rectangular amplitude, an experimental comparison was made. The results of this comparison are presented in Table IT' a t a single frequency together with the theoretical characteristics. The sinusoidal and rectangular data agree n-ell n-it,hin experimental error.
TIM E ( S E W
Figure 4.
Process transient response
The principal advantage of a quantitative treatment of automatic process control is the ability to predict desirable comhinations of cont,rol instrument,s and processes. Although there is no simple theoretical meane for predicting optimum adjustments of controllers, there are a number of empirical methods based on various dynamic characterizations of processes. Three of these methods were applied to the experimental data 1. The method of Ziegler and Kichols ( I O ) based on transient response 2. The method recommended by The Dvnaniic Systems Committee, Instrument,s and Regulars Divisio;, .4~;\113 ( 7 ) based on frequency response data 3 , The met,hod also of ziegler and Nichols based on the ultimate gain and ultimate frequency
Transient characteristics of the process were determilied from both positive and negative step pressure changes in the signal to the Moore regulator. The typical response curves are shown in Figure 4. The ultimate proportional band for the process, which is the maximum proportional band setting of the controller that will From the transient curves for the process (Figure 4 ) mean lead to sustained oscillation of the controlled variable, was found experimentally t o be approximately 4%. The Table IV. Comparison of Process Frequency Response Characteristics theoretical limiting band based on the computed frequency response characterRectangular Waves Sine Waves Theoretical 0 0830 0.0833 0,0830 0.0836 istics of the process and control]er is Frequency, radians/sec. 0.0838 0.0838 0.0836 Magnitude ra;io 0.0756 0.0765 0.0794 0.0853 0.096 0.0792 0.0792 1.27,. The respective ultimate frePhase angle, -127 -130 -132 -128.8 -133.G -131.5 -136 quencies were 1.4 and 2.6 cycles per ~
406
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 47, No. 3
ANALOGS response, since only the gain and frequency a t the ultimate condition of stability are needed. Ziegler and Nichols have related this method to the transient response method through the relations P , = 4L and
0 c a a
KP, = 8/5
W
Their recommendations for optimum control settings giving a '/r-amplitude recovery ratio for a step disturbanre for proportional control are PB = 2(PB),
n 3
t z
2 5 FREQUENCY
Figure 5. b
( RADIANS / S E C . )
Controller frequency response characteristics-magnitude ratio
for proportional plus reset control
PB = 2.22(PB), and reset rate
- Theoretical, sine wave
-__
T
=
1.2/P,
Theoretical, sine wave approximation
values of the slopes a t the point of inflection, and the time axis intercept of the tangent to the inflection point are inserted in the equations
I'B
=
SI,
for proportional control, and
The resulting control adjustments based on these three methods are shown in Table V for proportional control and proportional plus reset control. All the values listed are in terms of actual proportional bands and reset rates based on the calibratiods in Table I. The proportional band setting with reset obtained by the frequency response method is appreciably higher than that obtained by the other methods due to the fact that this method seeks stable control rather than optimum control.
P B = 1.1SL and R = 0 . 3 / L for proportional plus reset control. The input step magnitude applied in this case was 5 pounds per square inch. In terms of frequency response characteristics the rule of thumb requirement for stable, satisfactory control is t'lat 1. The open loop magnitude ratio or gain of the controlled process and its controller must not exceed 0.4 a t the frequencx for which the over-all phase angle, exclusive of the inherent 180 corrective angle in the controller, is 180" 2. The over-all phase angle should be less than 150.' a t the frequency for which the open loop magnitude ratio is unity
In terms of proportional band settings for proportional control alone this gain requirement corresponds to setting the proportional band a t not less than 2.5 times the ultimate proportional band. Over the important ranges of frequency for the analog the controller magnitude ratio is equal to the gain or reciprocal proportional band, and the phase angle above the inherent 180' corrective angle is zero. For proportional plus reset control the magnitude ratio log-log plot may be approximated by two straight lines, one of slope-1 beginning a t low frequencies, the FREQUENCY ( R A D l A N S l S E C ) other of zero slope a t high frequencies and equal to the proporFigure 6. Controller frequency response characteristional gain. The lines intersect a t a corner frequency in radians tics-phase angle per minute equal t o the reset rate in repeats per minute. The corresponding phase angles run from -90' a t low frequencies, Theoretical, sine wave; reset rate 2.50 repeats/min. through -45" a t the corner frequency, and approach zero a t high --- Theoretical, sine wave; reset rate 1.50 repeats/min. frequencies. Computed frequency response characteristics of the Bristol controller are shown in Figure 5 for settings of 100% The three methods used in predicting the controller settings actual proportional band and 2.5 repeats per minute reset rate. may be compared briefly. From the standpoint of experiment Dashed lines represent the straight line approximations. it is generally simpler t o obtain transient response data than In applying the frequency response method it is only necesfrequency response data. On the other hand a theoretical comsary to add together the characteristics of process and controller putation of transient response data is generally far more difficult for various controller settings until a combination is found that than a computation of frequency response data, because for the satisfies the recommended criteria. From the form of the conformer the differential equation must be solved, whereas for the troller curves and their relation to the corner frequency it is a simule matter to find a uractical combination. In the present case for purposes of comparison the reset rate was Table V. Recommended Controller Method Adjustments set a t the value found by the transient Transient Response Frequency Response Ultimate Gain Method response method. ProProProProportionalProportionalProportionalThe method of predicting controller portional reset portional reset portional reset adjustments based on the ultimat'e proProportionsl band, % 8.2 9.0 9.2 16 8.0 8.9 portional band is similar to but simpler Reset rate, reueatdmin. .. 1.5 .. 1.5 .. 1.7 than the method based on frequency
-
March 1955
INDUSTRIAL AND ENGINEERING CHEMISTRY
407
ENGINEERING, DESIGN, A N D PROCESS DEVELOPMENT latter a total solution is unnecessary. Furthermore it is a simple matter to determine over-all frequency response characteristics from component characteristics, whereas transient characteristics of components cannot be combined readily.
Summary The pneumatic process analog is a cheap and useful tool for studying automatic process control theory a t an elementary level. Its frequency response characteristics may be predicted reliably from the geometry of the eystem and from the basic rate and conservation equations for incompressible flow through capillaries between storage tank& The use of manually applied rectangular input Faves in measuring frequency response characteristics gives as accurate results as the use of pure sine waves a t low frequencies, thereby simplifying the apparatus requirements for frequency response analysis. The most convenient procedure for predicting controller adjustments for optimum regulation of simple processes like the analog is that based on the ultimate gain and ultimate frequency, both of which properties may be determined experimcntally or computed theoretically. For more complex processes frequency response methods offer the most practicable means and in many respects the only means of synthesizing stable control systems. The working principles involved in the application of frequency response techniques are simple, and since they are of great potential usefulness in the process industries, it is advantageous for chemical engineers to be familiar with them. Acknowledgment
zoo
100
TIME
( SECONDS
300
4(
1
Figure 7. Closed loop transient response at recommended settings
P
control + PR Proportional Proportional plus reset control
The ultimate gain or limiting proportional band and the C O I responding fiequency for a controlled process may be determined simply either by experiment or, by computation, if the process is describable by low-order linear differential equations. Of the three methods the one based on ultimate gain is the most useful for simple processes. Figure 7 shows the closed loop transient response curves obtained with 8% proportional band and with 9 % proportional band plus 1.6 repeats per minute reset rate. The effect of reset action is to eliminate any steady-state error (offset) but a t slightly reduced speed of response and an increased tendency to cycle. Theoretically the fractional offset v-ith 8% proportional band S/0.08) = 0.0174, which agrees with the obshould be S/(S served offPet of approximately 0.060. I n principle pneumatic analogs of the type studied heie may be used to represent specific processes which involve various combinations of simple RC stages. By choosing proper combinations of capillaries and tanks it is possible to match processes involving both cascaded and interacting stages. I n practice, however, the pneumatic analog lacks flexibility, since a continuous a i d e range of accurately known resistances and capacitances cannot be devised. Its utility is limited to simple RC-type systems, which generally can be simulated more conveniently by high speed electric analogs. Representation of systems having distributed rather than lumped parameters or having dead time and transportation lags is not readily attainable. On the affirmative side, the pneumatic analog is well suited for evaluating controller characteristics. Its real time behavior can be similar to typical chemical engineering operations making it particularly useful for studies of controlled systems in college courses on process control. It is inexpensive to build and operate, and, as has been shown, its behavior is predictable. In comparison with hydraulic analogs it is neater and more flexible; in comparison with electric analogs it is less expensive, less flexible, and involves processes more familiar to the chemical engineer.
+
408
This work vas supported in part by fundi: of the Eugene Higgins Trust allocated to Princeton University. The assistance of George Kopperl and Howard Zasloff in constructing the apparatus and malring preliminary studies is gratefully aclinom 1edged. Nomenclature = r0nFt:tnt = capacitance, 111. mahsllb. per sq. inch = capillary diameter, ft. = dimensional constant, 32.2 ft.-lb. mass/lb. force see.* = =
4-1
controller proportional gain, reciprocal proportional band fraction = capillary length, ft. = lag time on reaction curve, min. = ultimate period, min. = proportional band, = ultimate proportional band, yo = reset rate, repeats/min. = resistance, Ib./sq. inch sec./lb. mass = differential operator d / d t and root of characteristic equation = slope of ieaction curve a t point of inflection/unit input, % output change/niin. % ’ input = time sec. = time constant, RC, sec. = capacitance volume, cu. ft. = change in air density/unit change in pressure = absolute viscosity, Ib. mass/ft. see. = average air density, lb. mass/fL3 = process variable, pressure, Ib./sq. inch = frequencv, radians/sec.
r0
S t
T V
Ap 9 pa,
e
W
Literature Cited Aikman, A. R.,Trans. Am. Soc. Meeh. Enyrs., 76, 1313-23 (1954). (2) Ceaglske, N. H., and Eckman, D. P., IND. ENG.CHEM.,45, 1879 (1953). (3) Chestnut, H., and &layer, R. W., “Servomechanisms and Regulating System Design,” Wiley, New York, 1951. (4) Farrington, G . H., “Fundamentals of Automatic Control,’‘ Wiley, New York, 1951. (5) Johnson, E. F., ISD. ENG.CHEM.,43, 2708 (1951). (6) Mason, C. E., Trans. Am. SOC.Mech. Engrs., 60, 327 (1938). (7) Oldenburger, R., Ihid.,76, 1155-76 (1954). (8) Pessen, D. W., Ihid., 75, 843 (1953). (9) Rutherford, C. I., Proc. Inst. Mech. Engrs., 162, 334 (1950). (SO) Ziegler, J. G., and Nichols, X. B., Trans. Am. SOC.Mcch. E n g ~ s . 64, , 759 (1942). RECEIVED for review October 14, 19.54. ACCEPTEDJanuary 7, 1855. (1)
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 47, No. 3
