Automatic Control of a Pressure Process

---

ENGINEERING AND PROCESS DEVELOPMENT

Automatic Control of a Pressure Process N. H. CEAGLSKE'

AND

D. P. E C W N

Case lnstitute of Technology, Cleveland, Ohio

*

A

NALYSIS of automatic control characteristics of various thermodynamic and chemical processes is becoming necessary because of the reliance placed upon automatic control by the engineering designer. Processes are in use today in which economical operation is not possible without the various functions of instrumentation and automatic control. The trend toward catalytic reactions, high temperature, and high pressure is a part of the emphasis on greater reaction rate. I n addition the engineering designer tends to eliminate storage and holdup so that apparatus size can be reduced. These trends inevitably place a great burden on the automatic control system and require higher performance, greater speed, and better sensitivity of instrumentation equipment. This paper presents an analytical and experimental investigation of the dynamic characteristics of a simple gas pressure process involving thermodynamics and fluid flow. One theoretical method of analysis was employed: Frequency-phase response employing either theoretical or experimental values of process parameters ,

and a hydron (bellows) measuring element calibrated 0 t o 25 pounds per square inch gage. During the investigation reported in this paper the reset action of the controller was turned off. This process equipment is installed in the Instrumentation and Automatic Control Laboratory, Department of Mechanical Engineering, Case Institute of Technology.

A

Figure 1. Schematic Diagram of Process with Control PRC = Pressure recorder controller PC = Pressure regulator Fl = Flow indicator (rotameter)

The two experimental methods of analysis were: Process transient response to a step change Process frequency response to a sinusoidal change The various methods are compared as to accuracy and ease of performance in an industrial application. Process Equipment Performance Determines Proportional Control Settings

The process shown diagrammatically in Figure 1 consists of a pressure vessel exhausted to atmosphere through one to three nozzles. Air is supplied to the vessel from a high pressure source through a prwsure regulator and rontrol valve. The automatic controller is to maintain constant pressure in the vessel regardless of the magnitude of exhausting load on the system. Pressure Vessel. The vessel is a cylindrical closed tank of about l l a / , inches in diameter and about 2.40 feet long, with an accurately measured volume of 1.80 cubic feet. Exhaust Ports. The three exhaust openings consist of three 3/s-inch pipe caps drilled 0.089 inch. A globe-type shutoff valve is located in front of each opening. Control Value. The diaphragm control valve is a Conoflow Corp. Model AB10 with 1-inch valve body installed in a 1-inch line. The valve has nominal l/&xh ports. A valve positioner was not used. Flow Indicator, FI. The flow indicator is a Fischer and Porter rotameter especially calibrated for 13 standard cubic feet per minute maximum flow of air. Pressure Regulator, PC. The pressure regulator is a MasonNeilan Type 227, */4-inchsize. The regulator reduced the pressure from 35 pounds per square inch gage t o 25 pounds per square inch gage at the upstream side of the diaphragm control valve. Pressure Controller, PRC. The pressure controller is a Foxboro Co. Model 40 Stabilog with proportional plus reset action 1 Present address, Department of Chemical Engineering, University of Minnesota, Minneapolis, Minu.

September 1953

-

The analytical work consisted first of calculating the performances of the process equipment and then determining the best settings of proportional control. Process Analysis. I n any dynamical analysis of a thermodynamic process the investigator is confronted with the problem of whether the steady-state parameters of the process are adequate under conditions of automatic control-that is, under dynamical operation. In this paper the analysis is substantiated by experiment and it may be concluded that under the conditions of operation in these tests, the steady state parameters may be used. The process instrumentation diagram is shown in Figure 1 and the process operational diagram is shown in Figure 2. The dingrams are related as follows: 1. Set point

PRC.

R is the fixed setting of the pressure controller,

2. The error, E, is the difference between set point and indicated value of the controlled variable, B. 3. The controller, Go, including the error measuring means, is located in the pressure controller, PRC. 4. The variable, 21, of Figure 2 is the pressure output of the pressure controller, PRC. 5. The valve G,, is the diaphragm valve shown in Figure 1. 6. The variable, 2, is the valve top pressure fixing the rate of flow of air to the vessel. 7. The process, G,, is the vessel and outlet valves of Figure 1. 8. The controlled variable, C, is the pressure in the vessel. 9. The measuring element, G,, is located in the pressure controller, PRC. 10. The flow indicator, FL, plays no part in the closed loop of Figure 2. 11. The pressure regulator, PC, plays no part in the closed loop of Figure 2.

INDUSTRIAL AND ENGINEERING CHEMISTRY

1879

ENGINEERING AND PROCESS DEVELOPMENT Referring to the process in Figure 1,the law of conservation of mass is

c,-dP dt

= w1

- Wa

For operation in a small region near a given value downstream pressure and valve topworks pressure, the partial derivatives may be considered constant and Equation 8 becomes (9)

This equation employs the mass flow capacitance, C,, based on mass flow rate w.

and is where the constant of integration is taken aa zero because it has effect in calculating dynamic parameters, The process equation is obtained b y substituting Equations 6 and 9 into Equation 4, BO

CONTROLLER

VALVE

dP T -+P=K+

PROCESS E ssE L )

dt

(V

where the process time constant, T p , is ~

Figure 2.

V

T, =

M E A S U R I NGG M E L E M E N T

Schematic Representation of Closed Loop

and the process gain, K,, is

R = Set point of pressure controller y = Pressure output of pressure controller x = Valve top prenure C = Pressure in vessel B = Indicated value o f controlled variable E = Error, difference between R and B

The mass capacitance of the vessel may be calculated from the thermodpamic relation for a polytropic process,

P

- = constant

Equation 12 for process gain thus must take into account action of the control valve. Referring again to Figure 1, the valve lag is to be accounted for. Assuming that the valve top and tubing connection has only a single time constant, T,, then

Pn

and the definition of capacitance,

Combining Equations 1 and 2 there results

(3) This equation is based upon mass flow to. For volume flow a t standard conditions, the above equation is multiplied through by W l P OP 1, (4)

This equation is based upon the volunie flow a t standard conditions. Po. Thus the standard volume flovi capacitance is

where y is the output signal from the automatic controller (a pressure for a pneumatic controller). The automatic controller (Figure 1) is considered t o have proportional control and to act without lag of appreciable magnitude. Therefore, the control equation is y = K,e

(14)

with a proportional gain (inverse of proportional band) of Ke. The measuring element is assumed to have negligible lag and its equation is

b = c (15) The controlled system is now completely specified by Equations 10 for the process, 13 for the control valve, 14 for the controller, and 15 for the measuring element. Frequency Response Method of Analysis Requires Sinusoidal Signal

The outflon-. qo, depends upon the resistance of the exhaust nozzles, Po =

1

RP

because the downstream pressure at the nozzle is fixed (atmospheric). The resistance is

(7) where q is the standard volume flon-. The resistance a t a given operating region must be determined evperimentally because its calculation is hopelessly complicated by variable coefficients and a variable expansion factor in the usual gas flow equation. The inflow is a function of two variables, the downstream (vessel) pressure, P, and the control vaIve diaphragm pressure, x. If valve upstream pressure is considered constant,

The frequency-phase method of analysis requires introducing a sinusoidal signal (air pressure) a t E, Figure 2, and measuring the resulting periodic signal (air pressure) a t C or B (1-6). Once the system dynamical response is known, the amplitude ratios and phase lags may be calculated. I n the particular process under study, the controller has simple proportional action and the sinusoidal signal may be introduced a t point y, Figure 2, instead of point E. I n addition, it is difficult if not impossible to introduce a signal a t point E because of the construction of typical industrial pneumatic controllers. The object of this method is to determine the system response and then to calculate the best setting of proportional gain K,. The transfer function of each element of the system is defined by L (output) G ( s ) = L (input) and is the ratio of the Laplace t,ransforms of output and input at each element when initial conditions are all assumed to be zero.

1880

INDUSTRIAL AND ENGINEERING CHEMISTRY

Vol. 45, No. 9

ENGINEERING AND PROCESS DEVELOPMENT 1.0

The process transfer function is, from Equation 10,

-

The control valve transfer function is, from Equation 13,

The controller transfer function is, from Equation 14,

2

The measuring element transfer function has the value 1.0. The system transfer function is

3

01

ow

where only the controller gain, KO,is unknown. For single-loop control systems such as that of Figure 2, it has been shown (1.,.9)that good transient stability is obtained if K , is so adjusted that the phase lag from C to E is 120' to 150 when the amplitude ratio, C / E , is near the value 1.0, which corresponds to M P near 1.3 ( 1 , 2 ) . This simplified approach to the frequency response method (135' phase lag a t an amplitude ratio of 1.0) is justified in industrial process control for several reasons. The only method for calculation of control stability in common use is the ZieglerNichols (6) method which is sometimes very approximate. The method outlined above is an improvement over the ZieglerNichols method. I n view of variation in parameters., and with parasitic and essential nonlinearities., any calculation of control stability contains some error and an approximate single value of each controller parameter must be selected for use over a wide range of process parameters. In addition, it is doubtful if the mechanical engineer can be persuaded to use the frequency response method for industrial process control unless it is reduced t o the barest essentials as has been done in this paper. The frequency response of the system can be easily obtained by substituting iw for the Laplace operators in Equation 20. The resulting complex number is separated into ita real and imaginary parts, resulting in O

The phase of C/E then is the ratio of the negative imaginary part to the real part and is

01

FREQUENCY, CYCLES/SEC.

Figure 3.

sults. At this frequency, the amplitude ratio, iC/Ei-13,L, read from the graph, corresponds t o KpK, = 1. It is necessary now to determine the value of KpK, which will raise the curve so that the amplitude ratio will be one a t -135". This desired value of KpK, is the reciprocal of the value read from the graph n-ith KpK, = 1. For example, for the particular caw illustrated in Figure 3 the frequency a t 135' phase lag as calculated from Equation 22 is 0.108 and the amplitude ratio then as read from the graph is 0.0323. Therefore, for this case K P K , = 30.9.

FLOW, CU. FT./MIN. A T 14.7 LB./SQ. INCH A N D 70' F.

Figure 4.

The amplitude ratio or modulus of the complex number Equation 21 can also be calculated; however, the properties of the log-log A first-order transfer function such plot make this unnec-ry. 1

as -can be approximated (1, 9 ) by two straight lines interTs 1 secting a t WT = 1, the corner frequency. The first line from low frequencies up to the corner frequency will have an ordinate of 1. The second will hrtve a value of 1 a t the corner frequency and have a slope of -1. Then two first-order systems are assumed 1 These are to be combined so that C / E ( s ) = (1.45 1x26.45 1)' each plotted separately with the corner frequency for the first

+

+

being a t

w

=i

+

- or - = 0.714 radian per second or 0.114 cycle

T 1.4 per second. Likewise for the other part, the corner frequency is a t 0.006 cycle per second. T o obtain the combined curve the two individual curves are added together as shown in Figure 3. Assuming the desired phase lag is 135', the value of w can be calculated from Equation 22 or obtained from experimental reSeptember 1953

Frequency Response

Process Flow Characteristics

1. 2. 3.

Outlet valve No. 2 open Outlet valves Nos. 2 and 3 open Outlet valves Nos. 1, 2, and 3 open Upstream pressure 25 Ib./sq. inch

In some cases it may be desirable to specify M p rather than the phase lag for unity gain. The value of K , is then found graphically using a special plot (1,2 ) which is beyond the scope of this paper. Calculation, Transient, and Frequency Response Methods Evaluate Process Time Constant

The experimental work consisted of the determination of the process and control valve time constants by several methods. The response of the controlled system was aim tested. Calculated. The process capacitance (standard volume flow basis) is determined from the vessel volume, V , and the polytropic exponent, n, as in Equation 5. The volume of the tank and connecting piping was calculated from physical measurements and is 1.80 cubic feet. The polytropic exponent is determined by the expansion proc-

INDUSTRIAL AND ENGINEERING CHEMISTRY

1881

ENGINEERING AND PROCESS DEVELOPMENT

1 S

+

6

d

7

d

9

,O

//

FLOW, CU. FT./MIN AT 14.7 LB./SQ. INCH AND 70‘ F.

Figure 5.

Valve Flow Characteristics a t Constant Tank Pressure Indicated tank pressure, Ib./sq. inch Upstream pressure 2 5 Ib./sq. inch

ess of the gas (air) through the valve to the vessel. An n of 1.00 Jvould indicate isothermal behavior and an n of 1.41 would indicate adiabatic behavior. The procedure was to close the outlet valves and determine the time to reach a given pressure with the inflow rate held constant. The inflow rate was measured by a calibrated rotameter and the vessel pressure by the recording pressure gage. Then

The results for representative inflow rates are: Inflow Rate, Std. Cu. Ft./hIin. 4.5 7.5 10.5

Polytropic n 0.97 1.01 1.06

Thus the behavior of the process is nearly isothermal. As the process rate of reaction becomes higher, the exponent increases toward adiabatic behavior. At the speed of sound adiabatic behavior (n = 1.41) would be expected. The process resistance consists of the resistances of inlet and outlet valves. Equation 11 shows the equivalent resistance to be

The outlet resistance, ( bqo/bP1, M as determined experimentally by measuring inflow through the control valve at various vessel pressures and various control valvc openings. The result is shown in Figure 4. The outlet resistance is the slope of the curve at the given operating (vessel) pressure. T h e inlet resistance

(z)

was determined by measuring in-

flon- through the control valve a t various vessel pressures and various control valve openings. The results are shown in Figure 5.

Figure 7.

1882

Transient Test

TANK PRESSURE, LBJSCI. INCH.

Figure 6.

Valve Flow Characteristics at Constant Valve Pressure 1. 2.

Valve top pressure 10 Ib./sq. inch Valve tap pressure 5 Ib./sq. inch Upstream pressure 25 Ib./sq. inch

In this figure the inlet resistance is taken from the parameter (vessel pressure) changes with flow rate. However, this representation of the data is not convenient. The inlet resistance may be taken directly as a slope if vessel pressure is plotted against flow a t constant control valve pressure. Instead of cross plotting the original data, a new test was made and the results are shown in Figure G . The inlet resistance is the slope of these curves a t any given vessel pressure and valve top pressure. The values of resistances and capacitance are multiplied to obtain the process time constant, Tp. The results are shown in Table I. The process resistance, Re, varies considerably for various operating conditions. Table I.

Set Point, Lb./Sq. Inch 10.0 10.0 16.5 16.3 1&5

Calculated and Experimental Process Time Constant

ResistCapaciance, tame. Load Re. CW, Std. Cu. Lb. R.lin./ Ft.S/Lb. Ft./hlin. Ft.6 X 104 4.5 43 1 8.5 6.7 338 8 3 3.1 518 8.5 6.0 329 8.5 7.6 228 8.5

Process Time Conatant, TP = R e C V ,See. TranflreCalcd. sient quency 22.0 lQ.G 17.7 17.2 .. 26.4 .. .. 16.8 11.6 16:s ii:4

..

Transient Response. The process time constant was determined experimentally in order t o check the calculated value. First, a transient test \vas made by quickly changing the control valve position a small amount and recording the resulting rise in vessel pressure. The vessel pressure change was appro\imately 1 pound per square inch above the desired set point value. The pressure was recorded a-ith a Statham pressure pickup, Brush analyzer-amplifier, and Brush recording oscillograph, the equipment having good frequence response to 80 cycles per recond. The time constant is the time required to execute 63.2Yo of a total step change if the system is described by a firsborder differential equation. Remlts are shown by typical record of Figure 7 and summarized in Table I. The transient test values of the process time constant are in general agreement with calculated values. Frequency Response. A frequency response test was made by introducing a sinusoidal air prewure a t the control valve and recoding the vessel pres-

INDUSTRIAL AND ENGINEERING CHEMISTRY

VoI. 45, No. 9

ENGINEERING AND PROCESS DEVELOPMENT

Figure 8.

1

Frequency Test

sure as before. The sine wave generator was developed and designed a t the Case Instrumentation and Automatic Control Research Laboratory ( 8 ) . Typical records of input and output are s h o r n in Figure 8. The mean value of the valve top pressu'te was adjusted t o give the mean vessel pressure desired. The pressure recording equipment was calibrated under static conditions so that pressure differences could be read directly from the oscillograph chart. The resulting amplitude ratio and phase lag are shown in Figurea 9 and 10. The lower curve of Figure 9 shows test values at 10 pounds per square inch vessel pressure. Sine wave input amplitudes of 0.5, 1.0, and 1.5 pounds per square inch were used in order to determine any effect due to system nonlinearity. No significant difference can be noted since all test points appear randomly scattered. The intercept of the lower curve is a t 0.009 cycle per second and the time constant is calculated from 2nfZ'p = 1.0, then

Equation 12. The valve gain, ( b p i / b z ) ~is, the slope of the curves of Figure 5 a t any operating point. The results are shown in Table 11. Experimental (test) values were in close agreement. Valve and Measuring Element Time Constants. The valve time constant and measuring element time constrtnts were determined from a frequency response test by introducing a sinusoidal air pressure at point C, Figure 2, and measuring the resulting periodic pressure wave at point x; the latter is the valve topworks pressure, The observed amplitude ratio gives s valve time constant of 1.40 seconds and a measuring element time constant of 0.1 second.

Table II.

Set Point Lb./&. Inch 10.0 10.0 16.5 16.5 16.5 a

Tp =

1

--

=

17.7seconds

Experimental Process Constant

Gain and Valve Time

Load, Std. Cu. Ft./ Min.

Vessel Time Const., I'P, $eo.

Valve Time Comt.. Tv, Seo.

4.5 5.7 3.1 6.0 7.6

19.8" 17.2; 26.4 16.gb 11. 3 a

1.4 1.4 1.4 1.4 1.4

("i), &z

Process Gain, KP,

P

Ft.C/Lb. Min. x 104 41.7 52.2 24.2 53.5 59.8

Lb./Sq.

Inch per Lb ./ Sq Inch 1,87

.

1.79 1,34 1.82 1.40

Average of calculated and experimental values. Calculated.

(25)

2rf The upper curve of Figure 9 shows the process time constant at 16.5 pounds per square inch to be 11.4 seconds. These results are in general agreement with the calculated and transient values shown in Table I. The phase lags shown in Figure 10 do not necessarily need to be taken into account because a first-order linear system response is uniquely identified b y the amplitude response. However, the phase lags are in general agreement with the values calculated from the amplitude response (indicated by the solid lines of Figure 10). Considerable scattering of phase lag data ia common because phase lag is more difficult to measure than amplitude ratio. Process Gain from Characteristic Test. The process gain constant, Kp, may be determined from Figures 4 and 5 employing

Optimum Controller Gain Setting Depends upon Purpose of Process

The controller proportional band may be determined by the frequency response method a t a phase lag of 135 'or a t any specified Mp. The results are shown in Table 111. For the last condition of this particular system the phase lag of 135" corresponds to J f p of approximately 1.4. The proportional band may also be calculated for other values of Mp, as shown in Table IV. The final selection of the proportional band to be used depends not only upon the above but also upon the transient response and offset desired for the particular process. The lower the proportional band, the more oscillatory the response. A general statement applying to all processes cannot be made. Figure 11

d

F 4 u

o

I

01

0 01

FREQUENCY, CYCLES/SEC. FREQUENCY, CYCLES/SEC.

Figure 1.

9.

Frequency Response

Average tank pressure 10 Ib./sq. inch ' Lb./sq. inch to valve top 11.5 zk1.0

Figure 10.

80 2;:;

0 0

2.

September 1953

f0.5

Average tank pressure 16.5 Ib./sq. inch A 11 .O Ib./sq. inch to valve top

Frequency Phase Lag Curves

Theoretical for first-order systems 1. Average tank pressure 10 Ib./sq. inch Lb./sq. inch to valve top

&OS

2.

Average tank pressure 16.5 Ib./sq. inch A zk 1 .O Ib./sq. inch to valve top

INDUSTRIAL AND ENGINEERING CHEMISTRY

1883

ENGINEERING AND PROCESS DEVELOPMENT shows the experimental transient response for three proportional bands, 3.5, 6.3, and 8.3%. A4t4.4% the results would be between the first two. The problem may now be considered from two standpoints: What is the best value of controller gain for use on all five operating conditions of Table I11 What is the best value of controller gain for use a t each value of set point The first statement is made on the basis that the operating personnel would not desire to change the controller gain every time the load changed or every time the set point was changed. The second statement is made on the basis that the operating personnel may readjust the controller each time the set point was changed. It is a rare case when a controller is readjusted for every new value of load condition for a process.

Table 111.

Controller Gain

(Phasc lag 135')

Set Point Lb./Sd. Inch

Load, 3td. Cu. Ft./ Min.

Process Gain Lb./Sh. Inch per Lb./Sci. Inch

10 10 16.6 16.5 16.5

4.5 5.7 3.1 6.0 7.6

1.87 1.79 1.34 1.82 1.40

Calciilstrd Gain KC. lb./sq. inch per Over-all. Ib./sq. KPKC inch

Calcd. Propoy-

tional Rand", r70

12.9 12.2

3.73 3 . A3 2.08 21.1 4.14 ' 15.1 4.45 a Percentage of full range of controller required t o cause control valve t o c o r e r its entire range. I n this case the range of the instrument is 0 t o 25 lb./sq. inch a n d control valve opsrates between 3 and 15 lb./sq. inch. F o r first case, 1 12 1 12 Proportional hand = ~-X - 100 = -- X 7100 = 3 . i 3 ' i l K C 25 12.9 25

Table IV.

24.1 21.8

30.4

83.1 11.6 10.8

Controller Gain at Fixed Values of Mp

(Set point 16.5 lb./sq. inch, load 7 . 6 std.

OII.

ft./rnin.)

MP

1.3

15.1 10.6

1.2

9.5

1.4

Over-all 10 Ib./sq. inch set point 1 6 . 5 lb./sq. inch set point

10.8 7.6 6.78

4.4

6.3 7.1

drithnietic Mean 14.1 12.5 15.2

An exception would exist if there were different conditions surrounding the operation of the process a t diffprent wet point value.

Figure 1 1 .

Process Control Stability Tests

Set point 16.5 Ib./rq. inch, load 7.6 rtd. cu. ft./min. Proportional band, 4 3.5

5a 5b

Frequency Response Method Is Practical for Process Control

70

6.3

8.3

Best Single Value. h general criterion of selection is difficult t o state because the best value of gain depends upon the purpose which the process is t o accomplish and the method by which the process accomplishes the purpose. T h a t is, there is usually a preference between high overshoot and large damping, and low overshoot and small damping of a transient response t o a change of load because of conditions surrounding the manufacture of a product. Also, the process may spend more time a t one operating load than at another load. With proportional control it is usually desired to have the smallest possible offset. Assuming that overshoot and small damping are equally uridesirable and that the process operates about the same length of time a t each load, the arithmetic mean of values of proportional gain may be used, providing excessive (nearly continuous) oscillation does not result. Using the values of R,as calculated a t a phase lag of 135", the average is 14.1, equivalent to a proportional band of 3.4%. If this value were used for all cases the transient response would be slightly mole oscillatory for case3 1, 2, 4, and 5 , and less oscillatory for case 3. For process control, the use of the average controller gain is justified in niost cases; however, there may be special requirements in some particular processes in which it would be necessary to adjust the gain to give a better transient response. Best Value at Each Set Point. Xn inspection of the gain, K,, values of Table I11 shows that no better result is to be had by adjusting the band for each set point value because the mean values of gain a t the two set points are nearly the same:

1884

Process characteristics such as tinif: constants and gains must often be determined experimentally bccause their cnlculation is difficult, especially when fluid resistances are involved. Fluid capacitances are easily calculat,ed. The subject of fluid resietances would make a good topic for graduate engineering the.ws in either chemical or mechanical engineering. The transient method employcd in this paper is traditional. It has the great advantage of requiring little time and relatively little interruption of process duty. It has the great disadvantage of requiring very accurate measurement and very great sensitivity of the measuring element employed in recording the transient. This is true bemuse large dift'crences in process Characteristics may result in only slight differences in tyansient response. -41~0, in transient tests of processes, it is often difficult to niaintain all conditions constant except the two being teated. The frequency method has the great advantage of a l l o n k g accurate analysis from relatively crude measurements. ITo\r-ever, the time required for these tests ia relatively large when process time constant,s are 10 seconds a.nd larger. For example if a process has a time constant of T sccontl, a transient response, (one up, one d0w.n) can be run in about 1 i T seconds minilnuin time. A frequency response ( 2 c,ycles each of seven points) require 180T seconds minimum time. The frequency-phase method of adjusting gain (or other paraineters), for 135" phase lag a t an amplitude ratio of 1 or other values of M p , is simple and relatively foolproof for single-loop control systems. I n addition, i t has the advantage of allowing solution by simple graphical procedures involving only log-log coordinate paper and the use of a straightedge. It niuy be used for relat,ively high-order systems. However, the user should realize that this method does not set any other criterion for good response except that the system be relatively stable in transient

INDUSTRIAL AND ENGINEERING CHEMISTRY

Vol. 45, No. 9

ENGINEERING AND PROCESS DEVELOPMENT response and does not allow an optimum solution for a particular kind of disturbance of either set point (input) or load. The experimental and analytical work of this paper shows that the frequency response method of analysis is entirely practical for fluid processes of the usual kind encountered in chemical processing. Nonlinear dynamic behavior of such processes is not an important factor in analysis, so long as the solution is used in a reasonable operating range.

= output/input transfer function = process transfer function =

i KP

KO m

MP Steady State Calculations Agree with Transient Test Data

-r

The calculation of fluid process capacitance and resistance by employing values determined by steady-state calculation or test seems to be justified by the general agreement obtained in these tests between steady-state test data and transient calculations and test. This conclusion is logical because a discrepancy would not be expected until fluid velocities approached the order of magnitude of momentmuminterchange in the fluid or until gas velocities approach an appreciable Mach number.

n

P 5

Wi WO

X

Acknowledgment

X

The authors wish to express their appreciation to Case Institute of Technology and its Department of Mechanical Engineering and to the University of Minnesota for aponsoring this research while the first author was visiting professor at the former institution during the spring term, 1952. Thanks are also due George M. Lance, instructor in mechanical engineering a t Case Institute of Technology, for certain portions of the test work. Nomenclature = indicated value of pressure, lb./sq. ft. = controlled variable (pressure = P),Ib./sq. ft.

b c

Cf

= =

C, C,

=

e E

= =

f q

=

=

Laplace transform of c ( t ) mass flow capacitance, ft. sec.2 standard volume flow capacitance, fte6/lb. = cu, ft. per Ib./sq. ft. error (deviation), Ib./sq. f t . Laplace transform of e ( t ) frequency, cycles/sec. 32.2 ft./sec.2

u

Y Y*

P W

valve transfer function

= controller transfer function = constant = 4 Z-l

= process gain, Ib./sq. inch per lb./s inch = controller gain, Ib./sq. inch per %.,/sq. inch = Ib./ sq. inch in tank per lb./sq. inch on valve = mass, lb. sec.Z/ft. = maximum ratio [ C / R [ in the frequency range from 0 to = polytropic exponent = vessel pressure ( P = c), lb./sq. ft. = standard volume flow rate, standard cu. ft./sec. = fluid resistance, Ib. sec./ft.6 = Ib./sq. ft. per cu. ft./sec. = gas constant = Laplace transform operator = time sec. = absofute temperature, O R. or O K. = process time constant, sec. = valve time constant, sec. = vessel volume, cu. ft. = mass inflow rate, lb. sec./ft. = mass outflow rate, Ib. sec./ft. = valve top pressure, lb./sq. ft. = Laplace transform of x ( t ) = controller output pressure = Laplace transform of y ( t ) = error transfer function = density, lb./cu. ft. = frequency, radians/sec.

literature Cited

(1) Ahrendt, W. R., and Taplin, J. F., “Automatic Feedback Control,” New York, McGraw-Hill Book Co., 1951. (2) Brown, G. S., and Campbell, D. P., “Principles of Servomechanisms,” New York, John Wiley & Sons, 1948.

(3) Chestnut, H.,and Mayer, R. W., “Servomechanisms and Regulating System Design,” New York, John Wiley & Sons, 1951. (4) Eckman, D. P.,and Moise, J. C., presented a t the National Conference, Instrument Society of America, Cleveland, Ohio, September 1952. (5) Jamea, H.M., Nichols, N. B., and Phillips, R. S., “Theory of Servomechanisms,” New York, McGraw-Hill Book Co., 1947. (6) Ziegler, J. G . , and Nichols, N. B., Trans.Am. Soc. Mech. Engrs., 64, 8, 769 (1942). RECEIVED for review July 10, 1952.

A C C E P ~ EJune D 8, 1953.

Solute Transfer from Single Drops in Liquid-Liquid Extraction WILLIAM LICHT, JR., AND WILLIAM F. PANSING1 University of Cincinnati, Cincinnati 2 I , Ohio

T

0 DEVELOP further an understanding of the fundamental mechanism b y which solute is transferred during liquidliquid extraction in spray towers, an investigation was undertaken into the process of extraction from single drops passing through a stationary column of solvent. The basic premise, as proposed by Licht and Conway (7), was that in the life of each drop there must be three distinct stages and the mechanism of solute transfer must be studied separately in each stage. The stages are: 1

Present address, Standard Oil C o . (Indiana), Whiting, Ind.

September 1953

I. Drop formation-at

the nozzle or spray tip 11. Drop movement-through the column of continuous stationary solvent phase 111. Drop coalescence-at the interface a t terminal end of the column The motivation for this study came from an observation of Licht and Conway ( 7 ) regarding stage I, which seemed to be at variance with other results. Previous investigators working with single drops (8, 7 , 9, 11) have determined the amount of extraction during drop formation by making a plot of the logarithm fraction unextracted, or an equivalent variable, versus

INDUSTRIAL AND ENGINEERING CHEMISTRY

1885