10036_07e

712 CHEMICAL ENGINEERING

Figure 7.109 shows a simple ladder-logic diagram(@! This represents the logic statement (or pseudocode):

I F (X1 OR X2) AND X3 AND (NOT X4) THEN R1 = TRUE

This statement may be interpreted as-'if XI or X2 is true and X3 is true, but X4 is not, then output R1 is energised'. This is programmed into the PLC by pressing the appropriate keys on the keypad of the programming panel using the equivalent code (which in this case is for a Mitsubishi Melsec F1 series programmable controller-but is fairly typical):

LD XI OR X2 AND X3 AN1 X4 OUT R1

An important consequence of this method of programming is that the individual commands in the program are tested almost instantaneously and that the program cannot become locked into a small loop and thus cease to service any other control function.

7.22. REGULATORS AND ACTUATORS (CONTROLLERS AND CONTROL VALVES)

7.22.1. Electronic Controllers

Electronic modules are the industry standard for controllers employing a wide range of control strategies. Although, more recently, there has been rapid develop- ment of microprocessor-based controllers (see Sections 7.20 and 7.21) where control actions are simulated using software, hard wired systems+ based upon the integrated circuit (IC) and operational amplifier (op-amp) are still much in evidence.

The Operational Amplifier

equation 6.105, viz.: The output voltage V, of the op-amp shown in Fig. 7.1 10 can be expressed by

Vo = &mp(J'i, - FJ (equation 6.105)

where Ksmp is the gain of the amplifier.

FIG. 7.1 10. Operational amplifier

A hard wired system is one made up from a fixed set of electronic components whose characteristics cannot be changed by reprogramming software-if, indeed, software is available within the system.

Previous Page

PROCESS CONTROL 713

The properties of an opamp should approach the following as closely as possible'6'):

(a) Kamp should be infinite. (b) V, = 0 when Vi, = Vi2 (termed the common mode rejection ratio (CMRR)) . (c) The input impedance should be infinite. (d) The output impedance should be zero. (e) There should be no delay in the response of the output to a perturbation in input.

These properties give the general characteristics of operational amplifiers, viz.:

(i) Since the gain is infinite, then V, can only be finite if Vil = Vi2. (ii) As the input impedance is infinite, no current flows to the positive or

negative terminals of the amplifier. (iii) Since the output impedance is zero, an op-amp is able to provide the output

voltage V, whatever might be the impedance of the load connected to the output of the amplifier.

Proportional Action

From Fig. 7.1 1 1 and the general characteristics of the op-amp: Across resistor 1: 6, - V = iAR

Across resistor 3: V = i , R From equations 7.253 and 7.254:

Across resistor 2:

Across resistor 4:

From equations 7.256 and 7.257:

2 v = vi, h2 - V = iBR

V - V3 = iBR

2 v = vi2 + v,

(7.253)

(7.2 54)

(7.255)

(7.256)

(7.257)

(7.258)

/f/ FIG. 7.1 1 I . Use of the operational amplifier lo generate proportional action


From equations 7.255 and 7.258: V3 = vi, - Vi,

vo R , v3 R2

Also, from the proportional potentiometer:

_ - _ -

(7.259)

(7.260)

Hence, eliminating V3 between equations 7.259 and 7.260:

vo = - R , ( Vi, - Vi,) (7.261)

Thus, if Vi , represents the set point of the controller, Vi2 the measured value and R , / R 2 the proportional gain, then equation 7.261 is the equivalent of the relationship for proportional control action given by equation 7.2. The proportional potentiometer enables the proportional gain to be varied.

R2

PI Action

From Fig. 7.112:

Vi = iR, and V, - Vi = 1. i dt

where V, is the capacitance of the amplifier feedback circuit. %

. I (7.262)

If Vi corresponds to the error e, then the PI action relationships given by equations 7.6 and 7.262 are equivalent for unit proportional gain. R,%/ represents r, and the degree of integral action desired is controlled by adjusting R), i.e. the resistance of the integral action potentiometer.

Integral

potentiometer /action

FIG. 7.1 12. PI action using an operational amplifier

PROCESS CONTROL

%D i

71 5

potential

zero I volts -

volts ‘1 FIG. 7.1 13. Generation of PD action using an operational amplifier

PD Action

From Figure 7.1 13: 1 ‘

El 0 V ; = - 1 idt and V0 = -iRn

(7.263)

where FD is the capacitance of the input circuit. In this case, if V ; represents the error E , then equation 7.263 simulates derivative action

(Section 7.2.2) with FDR~, equivalent to TO. Varying RL) regulates the degree of derivative action desired.

PID Action

This is obtained by fitting together the modules illustrated in Figs. 7.1 1 1 , 7.1 12 and 7.1 13.

7.22.2 Pneumatic Controllers

For many years the pneumatic controller was preferred to its electronic counterpart due to its simplicity, its general ruggedness in the process environment, and the fact that its output could be used to operate directly the diaphragm of a pneumatic control valve. Although now largely superseded by software or hard wired electronic equivalents, pneumatic controllers are still employed in special circumstances, e.g. in explosive atmo- spheres. Furthermore, substantial numbers of pneumatic controllers can be found on older plant and thus an understanding of their principles of operation is necessary.

Proportional Con trol-Na rro w-Sand Action

Figure 7.1 14 shows how a pneumatic controller mechanism may be included in a simple control loop. The pneumatic output from the differential pressure cell


Orifice plate

Flow 1

L -- I

DP cell 1

restrictor IF Desired

FIG. 7.114. Arrangement of simple pneumatic flow control loop with narrow-band proportional control

(AP, = - A P ) increases with the flow of fluid through the orifice (see Volume 1, Section 6.3.3). This is transmitted to a recorder and controller, generally located adjacently in the plant control room. The pneumatic signal is converted into a mechanical movement by means of a spring and bellows device A. The bellows movement is linked both to the recorder pen and to the flapper of the controller mechanism which pivots about X. If the fluid flowrate in the line increases, AP, will rise, hence causing the flapper to move closer to the nozzle. This will restrict the flow of air through the nozzle and hence the pressure P upstream from the nozzle will rise. The latter is fed directly to the valve motor which acts in such a way that, as P increases, the valve shuts-thus reducing the flow in the line, and consequently AP,. With this type of controller mechanism the movement of the flapper required for P to vary over its entire range is very small (approx. 2 x l r ’m) . GOULD and SMITH@’) have shown that for such very small movements of the flapper, the change in output pressure AP is proportional to the movement of the flapper relative to the nozzle. The controller linkage is arranged so that the displacement of the flapper is proportional to the difference between the set point and the measured value AP,,

1.e.: AP a R - B = E (7.264)

(cf. equations 7.1 and 7.2). The extreme sensitivity of this system results in a very high proportional gain and, hence, is termed narrow-band proportional action-a high gain corresponding to a small or narrow proportional band (Section 7.2.3).

Proportional Control-Wide-Band Action

The excessive sensitivity of the narrow-band mechanism leads to considerable instability in the control system. This sensitivity is reduced by introducing a feedback bellows as illustrated in Fig. 7.1 15. In this case, if P increases due to the movement of the flapper towards the nozzle, the bellows will expand. Thus, as pivot Y is moved to the left by a change in measured value or set point, X will move to

PROCESS CONTROL 717

Spring loaded feedback bellows

Desired value

Measured value

FIG. 7.115. Pneumatic wide-band proportional controller mechanism

the right-so reducing the effective movement of the flapper relative to the nozzle. The resulting decrease in sensitivity is dependent on the extension of the bellows per unit change in P and on the ratio of 6, : 12. Such an arrangement can be to produce an action which approximates closely to equation 7.2 and proportional bands of up to 600 per cent can be obtained.

PI Action

integral action is added by the insertion of a restrictor and further bellows (Fig. 7.116). The rate of change of pressure in the integral bellows is proportional to the pressure driving force across the restrictor,

-

' 1.e.: -- dPI - X , ( P - P,) dt

SUPPI?

P

(7.265)

Desired value

Measured value

FIG. 7.1 16. Pneumatic generation of PI action

71 8 CHEMICAL ENGINEERING

Suppose, for t < 0, PI = P = Po, and that, at t = 0, Po is suddenly increased to Po + A P by the movement of the flapper, then initially:

(7.266) dPI dt - =.%,(Po + A P - P I ) = .%,(Po + A P - P o )

I=(

P I = .%i J=o A P d t

But, from equation 7.264, A P cx E . Hence:

(7.267)

P I = s% l' Edt (7.268)

Thus, the pressure of the output to the valve P is the sum of PI and the pressure produced by the wide-band proportional action contributed by the proportional bellows and the flapper-nozzle system (cf. equation 7.3). Note that for t > 0, P I is no longer equal to Po and thus equation 7.268 only strictly applies at t = 0. The value of 99 (and consequently TI) depends upon the capacity CIS of the integral bellows and the the resistance to flow RIR through the integral restrictor. It is generally assumed that C I S changes little and rI is varied by adjusting RIR.

PD Action

By inserting a restrictor in the line to the proportional bellows, any change in P will not be transmitted immediately to the feedback system. Thus, initially, the arrangement shown in Fig. 7. I17 will behave as a narrow-band proportional controller changing to wide-band action as the pressure in the feedback bellows PO approaches P . The rate at which P o + P depends upon the resistance to flow RDR through the derivative restrictor. This mechanism thus simulates derivative action in that it is most sensitive when the error is changing the most rapidly (Section 7.2.3). The derivative time q) is varied by adjusting RDR.

Proportional bellows

output to valve

Desired value

Measured value

FIG. 7-1 17. Pneumatic generation of PD action

PROCESS CONTROL 719

proportional Integral bellows bellows /

\ c /

Derivative restrictor Integral restrictor

supply --?= P

-

Desired value

value 9- Measured

T FIG. 7.118. Pneumatic generation of PID action

PID Action

and PD mechanisms as illustrated in Fig. 7.1 18. A pneumatic signal simulating PID action is obtained by a combination of the PI

7.22.3. The Control Valve

The control valve is a variable restriction in a pipeline which receives its position command from a controller-ither in the form of a single loop regulator or as part of a more complex control system. As such, the control valve constitutes by far the most common final control element although increasing use is being made of variable speed pumps and fluidics(64) to control the flowrates of process fluids.

A typical control valve is illustrated in Fig. 7.1 19. I t consists of three principal sections, viz.:

(i) The actuator or valve motor. This is the mechanism which converts the signal from the controller into the motion which positions the valve plug.

(ii) The valve body which contains the valve seat(s) and is the section of the valve through which the Auid passes.

(iii) The valve trim which consists of those sections of the valve in contact with the fluid which are detachable.

The Actuator

Figure 7.120 is a simplified view of a spring diaphragm actuator. The actuator receives a pneumatic signal from the controller via a boosterflow enlarger or a valve positioner and can be adapted in the form of an air-to-open or an air-to-close mechanism.

If the output from the controller is electric (e.g. 4-20 mA), then this can be converted to a pneumatic signal (with pressures up to 600 kN/m2) using an electropneumatic converter (Section 6.12.4). The resulting pneumatic signal then


,Actuator Nut for 8.

, Actuator stem

Coupling between actuator ’ and plug stem (also travel indicator) . Coupling and lock nut . Plug stem Threaded bushing Bonnet Guide bushing PTFE V ring-packing Spring

. Packing box Plug (with seat 03 38 mm plug guide) Seat Valve body

Valve positioner assembly

FIG. 7.119. Pneumatically operated control valve: (a) double-spring actuator with single-ported globe valve; (b) exterior view of double-ported control valve with

valve-positioner fitted on the side

PROCESS CUbJTROL 721

FIG. 7.120. Spring-diaphragm actuator: (a) air-to-open (direct) mechanism; (b) air-to- close (inverse) mechanism

Adjusting screws

Limit switch

Intermediate ch

Compression spring Compression spring

FIG. 7.121. Electrohydraulically operated actuator

Actuator body (oil r w o r ) Cover Control line Pilot Valve Contrd line Check valve Gear pump

Motor Cylinder Oil

Cyilnder housing

Piston rod Coupling Cup springs

Intermediate chamber

operates the actuator in the usual way. Alternatively, an electrohydraulic actuator can be employed (Fig. 7.121)(65) in each case, the actuator is constructed such that the displacement of the stem of the valve is proportional to the force applied by the control signal.


The Valve Positioner

Quite frequently the displacement of the valve stem will not be in proportion to the signal from the controller. This may be due to friction, an imbalance of forces on the valve plug(s) (particularly with single-seated valves) or to hysteresis in the spring/membrane actuator system@@. This difficulty can be overcome by the fitting of a valve positioner (Figs 7.122 and 7.1 196). The valve positioner is a high gain, mechanical-pneumatic feedback amplifier with gains.of the order of 10 : 1 to 100 : 1. The action of the mechanism is to compare continually the signal from the controller with the position of the valve stem (also represented as an air pressure) and to boost the pressure applied to the actuator membrane until the valve stem is in the correct position. Hence, the positioner also acts as a power relay as it amplifies the effect of any changes in pressure supplied from the controller.

Booster system

FIG. 7.122. Valve positioner fitted to a diaphragm actuator

The use of a valve positioner is generally beneficial with relatively slow control loops. Fitting a positioner within a fast control loop will decrease the stability of the loop(66).

Valve Body and Valve Trim

There are many different types of valve fitted with actuators to form control valves. Valves may be single- or double-ported (Fig. 7.123). With single-ported valves the valve plug is subjected to the total differential force across the valve. Such valves are sensitive to pressure fluctuations and powerful actuator elements are required for large pressure drops. Double-ported valves balance out the pressure differential but it is difficult to obtain complete shut-off.

PROCESS CDNTROL 723

Valve stem --.- - _ - Packing flange Packing Bonnet - Lubricator connection ---4__

Guide bushing Upper seat ring

Lower seat ring

Guide bushing

Valve plug

Bottom flange w n

(a) (b)

FIG. 7.123. Valve bodies showing typical trims used for control valves: (a) double-port globe valve with top and bottom guiding; (b) single-port globe valve with top and

bottom guiding

Valve bodies other than those illustrated in Fig. 7.123 which are employed for special-purpose control valves are ball valves, butterfly valves and Saunders diaphragm valves. These are described in Volume l , Section 3.5.4 and Volume 6, Section 5.3.

Control Valve Characteristics

The relation.between the valve stem position and the flow through the valve at constant pressure drop is termed the valve characteristic. Two characteristics must be evaluated for valve selection, the inherent and the installed characteristic^(^').

The inherent valve characteristic. This is a measure of the theoretical performance of the valve and is divided into:

(i) The decreasing-sensitivity type for which:

(ii) The linear (constant) sensitivity type where:

(iii) The increasing-sensitivity type for which:

(7.269)

(7.270)

(7.27 1)

where q is the valve-pow coeficient, Q is the volumetric flowrate of fluid passing through the valve expressed as a percentage of the maximum flowrate, and ts is the


a: Decreasing sensitivity type b: Linear (constant) sensitivity type c: Increasing sensitivity type

FIG. 7.124. Inherent valve characteristics

displacement of the valve stem expressed as a percentage of the stroke or valve lift (the maximum valve stem travel).

The different inherent characteristics (Fig. 7.124) are determined by the geometry of the valve orifice and valve plug and describe the volumetric flowrate through the valve as a function of the stem position with constant pressure drop across the valve.

Two other quantities which are important in sizing control valves are the turndown, which is the ratio of the normal maximum flowrate through the valve to the minimum controllable flowrate, and the rangeability, which is the ratio of the maximum controllable flowrate to the minimum controllable flowrate. Generally, valves are sized such that the turndown is approximately 70 per cent of the rangeability, i.e. that the maximum flowrate through the valve under normal operating conditions is about 70 per cent of the maximum possible flowrate. Typically, increasingsensitivity valves have higher rangeability values(6’). The installed valve characteristic. This depends upon the ratio of the pressure drop through the valve to the total pressure drop across the whole process line installation including the valve. If a valve of linear sensitivity is handling the entire system pressure drop, then its installed characteristic will also be linear. As the percentage of the pressure drop falls off, the installed characteristic rapidly changes to that of an on-off valve. The increasing-sensitivity valve also loses its characteristic in the same way, but to a far lesser extent. Hence, this type of valve is frequently chosen in preference to the linear type. PETERS(^') has discussed the effects of choosing valves with incorrect characteristics.

7.22.4. Intelligent Control Valves

The intelligent control valve system (ZCVS) is a recent innovation which differs from smart devices (see Section 6.12.5) in that it is a combined module which is able

PROCESS CONTROL 725

to perform all three major actions of a control loop (viz. measurement, control and process regulation) on a local basis(69). Thus, an ICVS possesses, at the least, sensors which monitor a number of process variables (e.g. fluid flowrate and temperature, upstream and downstream pressure, valve stem position, valve positioner characteristics, etc.) and a microprocessor of sufficient power to interpret these data, to communicate with a DCCS, and to be capable of acting as a control and self-calibration unit‘”). A distinct advantage of such a system lies in its ability to measure the flowrate of the fluid over the entire controllable range of the valve. This is because the pressure drop across the valve plug remains within its measurable range over the entire stroke of the valve stem due to the accompanying variation in the area for flow within the valve. A further feature of the ICVS is that it can be programmed to bring a process to a safe condition in the case of an emergency (e.g. loss of communication with an associated DCCS).

Intelligent control valves have found increasing application in processes which involve high-value materials and high labour costs such that their higher price can be justified.


7.23. APPENDICES

Appendix 7.1. Table of Laplace and z-Transforms

Description of time function Time function Laplace transform r-Transform -

f(t) f = [ zs ' f ( t )dr f(z)

Unit impulse &I) 1 1 1 - S Unit step U ( l )

Ramp

Quadratic

Exponential

Constant raised to the power r

Sinusoidal

Cosine

Multiplication by e-"'

Delay by time n 9

I

1 2

2 -

e- "1

sin wr

cos wr

z - 2 - 1

1 - s'

1 s + a -

2 - (0 0) 1 s - (1 / $ ) h a 2 - 0

W

s2 + w2

s2 + w2

S

z sin w 9 Z - 2 z c o s w 9 + 1

22 - 2 cos w 9 r 2 - 2 z c o s w 9 + 1

2 - 1 Zero-order hold element see Section 7.17.5 z

Exponential hold element 9 2

I + 8 s F=-F ,=w Bilinear transformation to map the interior of the unit circle in the r-plane onto

the left half of the complex variable {-plane. (Application of the Routh-Humitz stability criterion). c - 1

In the above 9 indicates the sampling interval.

Appendix 7.2. Determination of the Step Response Function of a Second-Order System from its Transfer Function

From equation 7.81:

where:

.. (7.A2.1)

PROCESS CONTROL

Inversion of equation 7.A2. I gives:

where # = 5’( I - C’) is real.

..

Similarly:

727

(7.A2.2)

(7.A2.3)

(7.A2.4)

(7.A2.5)

From equation 7.A2.1 by partial fractions: I

A=m i.e.:

.. 1 Also: E =

A(Pl - 82) 1.e.:

Substituting from equation 7.A2.3:

..

Similarly:

22

2#(# - iC) - - -

Substituting for A, E and C in equation 7.A2.3 and using equations 7.A2.4 and 7.A2.5:

- MKMT I j = I - ~ x p [ - j l ) ( ~ c O s - l + - 0 r #’+ 9‘ (2 s i n 2 r I)


But:

and:

5 )sin- r (7.A2.6) ‘ 1 - 1 8 = 1 - exp[- f t]b- ’ r + - .. MKMT -/(I - c’

Equation 7.A2.6 can be put into a more useful form by using the trigonometric identity: pcos a+ q sin a= rsin (a + p){

where: f = d(p2 + 4) and: tan 4 =

9

Thus:

Thus equation 7.A2.6 becomes:

A*! r r

where Y = .I({’ - 1) and is real. As for c c 1, from equation 7.A2.1 by partial fractions:

A = r 2

1 8I(PI - 82) B =

.. ’ B = - 7’ and:

2 v ( v - 0

(equation 7.82)

52 Similarly: C=- 2v(v+ (?

Substituting for A, B and C in equation 7.A2.2:

12 MKM,

1 = I - exp[- $)[- - I exp[- :I (7.A2.7) MKMT

Thus:

cosh-Sf[exp[?)+exp(-:] r Putting:

and:

PROCESS CONTROL 729

in equation 7.A2.7, leads to:

8 = MKM, { ( : I f I - cosh - + - sinh y']exp 7 [- 6) (4 c = I

Under these circumstances:

Thus eauation 7.A2.1 becomes:

= A + B + - C s (s+ 1/02 s + l / z

Equating coefficients etc. gives: A = r , B = - z and C = - z'

MKMT s (s + IItj' s + I / r - 1 g=-------.-- I I / r 1 ..

Inverting equation 7.A2.8:

(equation 7.83)

(7.A2.8)

(equation 7.84)

7.24.. FURTHER READING ASTROM; K. J. and WITTENMARK, B.: Aabptive Control (Addison-Wesley, Wokingham, U.K., 1989). BENNETT. S.: Real-Time Computer Control: An Introduction (Prentice-Hall, Hemel Hempstead, U.K.,

COUGHANOWR, D. R.: Process Systems Analysis and Control, 2nd edn. (McGraw-Hill, New York, '1991). KUO, B. C.: Discrete Data Control Systems (Prentice-Hall, Englewood Cliffs, New Jersey, 1970). LANDAU, Y. D.: Adaptive Control-The Model Reference Approach (Marcel Dekker, New York. 1979). POPOVIC. D. and BHATKAR, V. P.: Distributed Computer Control for Industrial Automation (Marcel

RAVEN, F. H.: Automatic Conrrol Engineering, 2nd edn. (McGraw-Hill, New York, 1968). SHINSKEY, F. G.: Dbtillation Control (McGraw-Hill, New York, 1977). STEPHANOPOULOS, G.: Chemical Process Control: An Introduction to Theory and Practice (Prentice-Hall.

WARNOCK, I. G.: Programmable Controllers (Prentice-Hall, Hemel Hempstead. U.K., 1988).

1988).

Dekker, New York, 1990).

Englewood Cliffs, New Jersey, 1984).

I .

2. 3. 4. 5 .

6.

7.

8.

7.25. REFERENCES BS 1646: 1979: BRITISH STANDARD 1646 (British Standards Institution, London) Symbolic repres- entation for process measurement control functions and instrumentation. PETERS, J. C.: Trans. A.S.M.E. 64 (1942) 247. Experimental studies of automatic control. SHINSKEY, F. G.: Distillotion Control (McGraw-Hill, New York, 1979). PARKINS. R.: Chem. Eng. Prog. 55 (7) (1959) 60. Continuous distillation plant controls. MURRILL, P. W.: Automatic Control of Processes (International Textbook Company, Scranton, Pennsylvania, 1967). STEPHANOPOULOS, G.: Chemical Process Control: An Introduction to Theory and Practice (Prentice- Hall, Englewsod Cliffs, New Jersey, 1984). ACTON, J. R. and SQUIRE, P. T.: Solving Equations with Physical Understanding (Adam Hilger Ltd. Bristol, 1985). DEL TORO, V. and PARKER, S. R.: Principles of Control Systems Engineeiing (McGraw-Hill, New York, 1960).


9.

10.

11. 12.

13. 14.

15.

16. 17. 18.

19.

20. 21. 22.

23. 24.

25.

26.

27.

28.

29.

30. 31. 32.

33. 34. 35. 36.

37.

38.

39. 40. 41. 42.

43. 44. 45.

46. 47.

48.

BRITTON, J. R., KRIEGH, R. B. and RUTLAND, L. W.: University Murhemurics. Vol. 2 (Freeman, San Francisco, 1965). CRANDALL, E. D. and STEVENS, W. F.: A.I.Ch. E.JI. 11 (1965) 930. An application of adaptive control to a continuous stirred tank reactor. DOUGLAS, J. M.: Process Dynumics and Conrrol (Prentice-Hall, Englewood Cliffs, New Jersey, 1972). ROSE, A. and WILLIAMS, T. J.: Ind. Eng. Chem. 47 (1955) 2284. Automatic control in continuous distillation. HARRIOT, P.: Chem. f i g . Prog. 60 (8) (1964) 81. Theoretical analysis of components. BENNETT, S.: Red-Time Computer Conrrol: An Inrroducrion (Prentice-Hall, Hemel Hempstead, U.K., 1988). MICKLEY, H. S., SHERWOOD, T. K. and REED, C. E.: Applied Murhemurics in Chemicul Engineering (McGraw-Hill, New York, 1957). CEAGLSKE, N. H.: Auromuric Process Controlfor Chemicul Engineers (Wiley, New York, 1956). WYLIE, C. R.: Advunced Engineering Murhemurics (McGraw-Hill, New York, 1966). COUGHANOWR, D. R.: Process Sysrems Analysis und Conrrol, 2nd edn. (McGraw-Hill, New York, 1991). HURWITZ, A.: Marh. Annln. 46 (1895) 273. Obcr die Bdingungcn, unter welchen cine Gleichung nur Wurzeln mit negativen realen Teilen bcsitzt. ROUTH, E. J.: Advunced Rigid Dynamics (Macmillan, London. 1884). NYQUIST, H.: Bell System Tech. J. 11 (1932) 126. Regeneration theory. BODE, H. W.: Bell Sysrem Tech. J. 19 (1940) 421. Relations between attenuation and phase in feedback amplifier design. TAKAHASHI, T.: Murhemurics of Auromuric Conrrol (Holt, Rinehart and Winston, New York, 1966). BODE, H. W.: Network Analysis und Feedbuck Amplifier Design (van Nostrand Reinhold. New York, 1945). JAMES, H. M., NICHOLS, N. B. and PHILLIPS, R. S . : Theory ofServomechanhs (McGraw-Hill, New York, 1947). ZIEGLER. J. G. and NICHOLS, N. G.: Truns. A.S.M.E. 64 (1942) 759. Optimum settings for automatic controllers. COHEN, G. H. and COON, G. A.: Truns. A.S. M . E. 75 (1953) 827. Theoretical consideration of retarded control. LOPEZ, A. M., SMITH, C. L. and MURRILL, P. W.: Brit Chem. f i g . 14 (1969) 1953. An advanced tuning method. EDGAR, T. F. and HIMMELBLAU, D. M.: Optimizurion of Chemicul Processes (McGraw-Hill, New York, 1988). SMITH, 0. J. M.: Chem. Eng. Prog. 53 (5) (1957) 217. Close control of loops with dead time. RAVEN, F. H.: Aufomufic Conrrol Engineering, 2nd edn. (McGraw-Hill, New York, 1968). WARDLE, A. P. and WOOD, R. M.: I. Chem. EJEuropeun Federation of Chem. Eng. Inr. Symp. on Disrillurion (1969) Session VI, 66. Problems of application of theoretical feed-forward control models to industrial scale fractionating plant. SHINSKEY, F. G.: Process Control Systems, 2nd edn. (McGraw-Hill, New York, 1979). SHINSKEY. F. 0.: Disrillurion Conrrol (McGraw-Hill, New York, 1977). ROSENBROCK, H. H.: Computer-Aided Conrrol System Design (Academic Press, New York, 1974). BRISTOL, E. H.: IEEE Truns. Aurom. Conrrol AC-11 (1966) 133. On a new measure of interaction for multivariable process control. STATHAKI, A., MILLICHAMP. D. A. and SEBORG, D. E.: Can. J. Chem. Eng. 63 (1985) 510. Dynamic simulation of a multicomponent distillation column with asymmetric dynamics. KOCHENBURGER, R. J.: Truns. AIEE 69 (1950) 270. A frequency response method for analysing and synthesising contactor servomechanisms. TRUXAL, J. 0.: Auromuric Feedbuck Conrrol Sysrem Synthesis (McGraw-Hill, New York, 1955). GRAHAM, D. and MCRUER, D.: Anulysir of Non-Lineur Conrrol Sysrems (Wiley, New York, 1961). Kuo, B. C.: Discrete Daru Conrrol Sysrems (Prentice-Hall, Englewood Cliffs. New Jersey, 1970). LUYBEN, W. L.: Process Modeling, Simulurion und Conrrol for Chemicul Ihgineers (McGraw-Hill, New York, 1973). YOUNG, R. E.: Supervisory Remore Conrrol Systems (Peter Peregrinus, Stevenage. 1977). DAHLIN. E. B.: Insrun . Conrrol Sysr. 41 (6) (1968) 77. Designing and tuning digital controllers. KALMAN. R. W.: Truns. ASME 80 (Series D) (1958) 468. The design of a self-optimising control system. ASTR~M, K. J. and WITTENMARK, B.: Adaprive Conrrol (Addison-Wesky, Wokingham. U.K., 1989). MORANT, F., MARTINEZ, M. and PIC^, J.: In Application of Arri~ciul Intelligence in Process Conrrol by BOULLART, L., KRIJGSMAN, A. and VINGERHOEDS, R. A. eds, Section VI. Supervised adaptive control. LANDAU, Y. D.: Adaptive Conrrol-The Model Reference Approuch (Marcel Dekker, New York, 1979).

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CLARKE, D. W. and GAWTHROP, P. J.: IEE Proc. 122 (9) (1975) 929. Self-tuning controller. CLARKE, D. W., MoHTm!, C. and TUFFS, P. S.: Auromafica 23 (1987). Generalised predictive control: parts (i) and (ii). Pomvrc, D. and BHATKAR, V . P.: Distributed Computer Control for Industrial Automation (Marcel Dekker. New York, 1990). CIVERA, P.. DEL CORSO, D. and GREGORETT!, F.: In Microprocessors in Signal Processing, Measure- ment and Control, TZAFESTAS, S. G. ed. (Reidel, London, 1983). Microcomputer systems in real-time applications. JOVIC, F.: Expert Systems (Kogan Page, London, 1987). BAILEY, R. W.: Human Perjormance Engineering: A Guide for System Designers (Prentice-Hall, Englewood Cliffs, New Jersey, 1982). GILMORE, W. E.. GERTMAN. D. 1. and BLACKMAN, H. S.: User-Compurer Inferface in Process Control (Academic Press, London, 1989). KALANI, G.: Microprocessor Based Disrribured Control Systems (Prentice-Hall, Englewood Cliffs, New Jersey, 1988). CAUNT, E.: Process Industry Journol (January, 1994). It’s the application that’s important. WARNOCK, I. G.: Programmable Conrrollers (Prentice-Hall, Hemel Hempstead, U.K., 1988). Mitsubishi Electric Corporation: Instruction Manual HI-IE-041-G (1992). The Melsec-F programmable controller. JONES, M. G. and WARDLE, A. P.: Proceedings of rhe first IASTED Internarional Symposium on Circuits and Systems, Zurich (1991) 90. Design of a PLC-based control system for a batch reactor. SMITH, J. I.: Modern Operational Amplifier Design (Wiley, New York, 1971). GOULD, L. A. and SMITH, P. E.: Instr. 26 (1953) 1026. Dynamic behaviour of pneumatically operated control equipment (Part 11). COUGHANOWR, D. R. and KOPPEL, L. B.: Process Systems Analysis and Control (McGraw-Hill, New

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A

York, 1965). SINGH. J.: Process Indusrrv Journal [January 1993) 46. Look no movina oarts. Samson plc: Data sheet T834UE (Makh, 1992). Electrohydraulic activato; for type 240 control valves. SINGH, M. G., ELLOY, J. P., MEZENCEV, R. and MUNRO, N.: Applied Industrial Control, Vol. 1 (Pergamon Press, Oxford, 1980). WHERRY, T. C., PEEBLES, J. R., MCNEESE, P. M., WORSHAM, R. E. and YOUNG, R. M.: In PerryS Chemical Engineers’ Handbook, GREEN, D. W. and MALONEY, J. 0. eds., 6th edn. (McGraw-Hill, New York, 1984). Section 22, Process control. PETERS, J. C.: Ind. Eng. Chem. 33 (1941) 1095. Getting the most from automatic control. WHITE, 9. A.: Chem. Eng. Prog. 89 (12) (1993) 31. Rethink the role of control valves. COURT, R.: Process Industry Journal (June 1993) 58. Actuators are getting smarter.

7.26. NOMENCLATURE Units in SI Dimensions System in M, N, L,

T, 6 A Cross-sectional area Mean areas for heat transfer A , , A2

A,, A, Constants Al Area under pulse function ALM Log[l/(gain margin)] E El, E2, etc. Constants b j I , b32, etc. Elements in Routh-Hurwitz array C Controlled variable C1, C2 Average specific heats C,B C+ ‘&I

’&o

c , , c, - I , etc. Elements in Routh-Hurwitz array D D Decoupling element

Measured value of a variable

Capacity of integral action bellows Deviation in controlled variable due to interaction Capacitance of amplifier feedback circuit in hard-wired

Capacitance of voltage input circuit in hard-wired system

Overhead product flowrate in mass or moles per unit time

system generating PI action

generating PD action

m2 m2

F (farad)

F (farad)

kg/s, kmol/s -


Units in SI Dimensions System in M, N, L,

T, e. A d Diameter of manometer tube m L F Feed flowrate in mass or mols per unit time kg/s, kmol/s MT-’, NT-’ f Function -

z-transform of function - (Cyclic) frequency Hz Transfer function - C Transfer function of final control element GI

G2, C; Transfer function of process Cc Transfer function of controller - GC(hri,, Transfer function of industrial controller derivative module - Ccomp Transfer function of lead, lag or lag-lead compensator - CDrc Transfer function of dead time compensator - cest C F F Cil

C,I CX CY

GO)

9” - -

CZOH

Transfer function assumed by estimator (equation 7.252) Feed-forward controller transfer function Transfer function of inner loop Transfer function of primary closed-loop system Transfer function of secondary closed-loop system Transfer function of blocks in series Transfer function of blocks in parallel Transfer function of zero order hold element Transfer function of discrete time system, i.e. in terms of

Pulse transfer function (defined in equation 7.210) Acceleration due to gravity Transfer function of measuring element or of elements in

Heat transfer coefficients

Current Controller output signal a t time t Controller output signal for zero error Number of sampling periods equivalent to the system

dead-time in Dahlin’s algorithm Steady-state gain Flow-head relationship in equations 7.21 and 7.40

respectively Flow-head relationship in equations 7.25 and 7.29

respectively Gain of operational amplifier Proportional gain Constant in equation 7.4 Constant in equation 7.3 Magnitude of veztor at the frequency where polar plot

cuts negative real axis Steady-state gain for manometer tube Steady-state gain of process reaction curve

(in Cohen-Coon procedure) Ultimate gain Constants in equations 7.265-7.268 Input/output relationship for linear section of non - I’ inear

the z-transform

feedback path of control loop

4- 1)

characteristic LI, LR e

L2, LN- I , LF Liquid flowrates in mass or moles per unit time

e2

kg/s, kmol/s kg/s, kmol/s m

controller m - of the stroke

Reflux flowrate in mass or moles per unit time Length of liquid column in manometer tube Distances measured along flapper of pneumatic

Distance of valve stem travel expressed as a percentage e, M Magnitude or amplitude of signal -

PROCESS CONTROL 733

Dimensions in M, N, L, T, 8, A

Mlr mz N N n

PZ

Qo QI, QZ Q. QH

q n

R

RD RDR RI

r

S S

RIR

S

I 10 U

Vi

Manipulated variable Change in manipulated variable required to compensate

Mass Number of stages or total number of samples Describing function Index giving type of polar plot or number of samples or

Net number of encirclements of the point (- I , 0) on the

Number of independent equations Number of degrees of freedom Number of independent variables Number of zeros of function Pressure upstream of nozzle in flapperlnozzle system Pressures applied to limbs of manometer tube or pressures

downstream and upstream of orifice plate Distillation column pressure Pressure in feedback bellows of pneumatic controller Frictional drag per unit cross-sectional area of

Pressure in integral bellows of pneumatic controller Output pressure from differential pressure cell Volumetric flowrate or volumetric flowrate expressed as a percentage of

maximum flowrate Volumetric inlet flowrate Volumetric outlet flowrate Heat per unit time supplied to boiler Heat per unit time supplied by flowing stream Heat to vaporise one mole of feed divided by molar latent

Heat to vaporise one mole of reflux divided by molar

Set point or desired value of a variable or shear stress on surface or electrical resistance Resistance of derivative action potentiometer Resistance of derivative restrictor Resistance of integral action potentiometer Resistance of integral restrictor Number of roots of characteristic equation or number of

poles of closed-loop transfer function Steam flowrate to reboiler in mass or mole per unit time Sum Laplace transform parameter Ultimate period Sampling time Time Duration of pulse function Load variabie(s) Unit step function Mean velocity Vapour flowrate in mass or moles per unit time or voltage Input voltage for operational amplifier Output voltage for operational amplifiers Volume of liquid in tank

for interaction

number of terms in Fourier series

complex plane

manometer tube

heat

latent heat

Units in SI System

- N/m2 N /m2

N/m2 N/mz

N /m2 N/m2 Nlm2 m’/s

- m3/s m’/s W W

- - N/mz n n s/m3 n s/m3

- kg/s, kmolls

1 /s -

S S S S -

m/s

kg/s, kmol/s V V V m3

- M

- - ML-~T-’ M L - ~ T - ~

M L - ~ T - ~ M L - ~ T - ~

M L - ~ T - ~ ML- I T - ML- I T - L’T-’

ML- I T - MLzT-3A-2 MLzT-3A-2

- MT-’, N T - l

T-1 T T T T

-

- - LT-I

CHEMICAL ENGINEERING 734

VN W

X

X u

X D

Xrn x w

W

X

X F

X

yo Y Y Z

Z a

Vector in the complex plane Bottoms product flowrate in mass or moles per unit time Mass flowrate Output of dead time compensator Input to control loop component Composition of stream recorded by analyser (mole or

Molar mass fraction of component in overhead product Molar mass fraction of component in feed Composition of main stream (mole or mass fraction) Molar mass fraction of component in bottoms product x dimension Magnitude of on-off signal (equation 7.194) Output from control loop component y dimension Liquid head or z-transform parameter z dimension Angle

mass fraction)

.. al , a2, etc. Roots of the characteristic equation (poles of the

transfer function) sin-’(6/M) (Section 7.16.2) Real parts.of the roots of the characteristic equation or real

parts of the complex variable s Imaginary parts of the roots of the characteristic

equation or imaginary parts of the complex variable s Magnitude of dead-zone Dirac function Error Difference between output of reference model and

process output Damping coefficient Valve-Row coefficient Temperature Measured temperature Set point value of temperature Interaction factor Relative gain array Relative gain Viscosity of fluid

Coefficient in Fourier Series Complex variable defined in equation 7.230 Density Linear range of element with saturation Parameters (constants) in equation 7.139 and Table 7.5 Time constant Apparent time constant Apparent dead time Distance-velocity lag, dead time Derivative time Integral time

$c2 - 1)

Tunyng parameter in equations 7.139-7.142

Phase shift 4 1 - t2) Phase shift of lead, lag or lag-lead compensator Angle representing phase margin on Nyquist diagram Angular frequency Comer or break frequency


T, 8, A

k g h , kmol/s MT-I, NT-’ M T - I

PROCESS CONTROL 735


T, 8, A

T-' (4co Cross-over frequency I Is T-'

9 1 ~ Real part of function - - gn, Imaginary part of function - -

- Sampling frequency 1 Is - cu, Re Reynolds number with respect to pipe diameter

Primed symbols represent the steady-state value of the variable (e.g. 0').

Symbols in script (unless listed specifically in the nomenclature) denote the deviation from the steady-state value of the variable (e,g. 9 = 6 - 0'). Symbols with bar represent the Laplace transform of the variable (e.g. 8 i s the Laplace transform of 0).

Starred symbols represent a sampled output (e.g. To(!) is the sampled data equivalent of the continuous signal f(t)). Prefixes: A represents a finite change in a variable.

Acronyms

N D AR (AR),, CCR CM CMRR CPU Dl A DCCS DDC DH DM

El P FRC GMV GPC HFA IAE IC ICVS I10 ISE ITAE LAN LC LCR LED LFA LIC MBC MIMO MRAC 0 s P PD PI PID PLC PM RAM

D V ( W

Analog-to-digital Amplitude ratio Amplitude ratio at the cross-over frequency Central control room Control module Common mode rejection ratio Central processing unit Digital-to-analog Distributed computer control system(s) Direct digital control Data highway Data acquisition module Distance-velocity (lag), dead time Elect ropneuma t ic Flow recorder controller Generalised minimum variance controller Generalised predictive controller High frequency asymptote Integral of the absolute error Integrated circuit Intelligent control valve system Inputloutput Integral square of the error Integral of the time-weighted absolute error Local area network Level controller Local control room Light emitting diode Low frequency asymptote Level indicator controller Microprocessor-based control Multiple-input/multiple-output Model reference adaptive controller Operator Station Proportional control action Proportional plus derivative control action Proportional plus integral control action Proportional plus integral plus derivative control action Programmable controller (programmable logic controller) Phase margin Read and write memory

736 RCM RDM RGA SCADA SS R STR TRC TTL ZOH

CHEMICAL ENGINEERING

Reserve control module Reserve data acquisition module Relative gain array Supervisory control and data acquisition Solid-state relay Self-tuning regulator Temperature recorder controller Transistor-transistor logic Zero order hold element

10036_07e

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