M. A. Larron and 0. A. ~ e n g ' Iowa State University
I
Process Dynamics Experiment
Chemical processes are becoming more complex and consequently must be designed with extensive and complicated control systems. Graduating chemical engineers and graduating chemists interested in chemical processing should have at least a rudimentary understanding of the principles behind process control design. A process control system is composed of a t least the following elements: controller, final control element (e.g., control valve), process, measuring device and mechanism t,o compare the value of a measured variable with its desired value. The dynamic analysis of process control systems encompasses the study of the interaction of these elements. Time dependent differential equations, relating information output to information input, are developed for each element. These equations are subsequently combined to give an equation representing the dependency of some process variable on a system upset. For example, the variable may be temperature, pressure, liquid level, pH, or concentration. Possible system upsets are change in controller setting, change in flow rate to or from the process, change in energy input, or change in ambient temperature. The validity of the equations must Grst be verified. They can then he used to determine control design, process design, and system stability. The two most common methods of utilizing these equations are transient analysis and frequency analysis. For transient analysis the system or element of a system is subjected to a "step" change; that is, some input (control setting, flow rate, etc.) is instantaneously changed by a measured amount. The response of the measured variable is observed and compared to that predicted by the equation. Methods for using experimental transient data for control system analysis are described by Ziegler and Nichols (1). In the frequency analysis of a system, some input is varied periodically in time. The value of the measured variable becomes a periodic fuiiction of time and is, in general, out of phase with the input signal. The amplitude and phase lag or lead of the output signal are functions of frequency and the parameters of the system. Techniques for utilizing this information are described in standard tests in process control (2, 3). Because of the complexity of most control systems, the solution of the differential equations must be carried out on a computer. The analog computer lends itself readily to the solution of ordinary linear differential equations of the type generated by control
1 National Science Foundation Fellow, 1960-61.
systems (4). Entire control systems can be easily represented by analog computer simulation. Parameters in control and process design equations can be reduced to dial settings, consequently the response to many designs can be obtained in a relatively short time. Much has been published demonstrating the techniques of control analysis and computer simulation. Some of the published work discusses pH control ( 5 ) , heat exchanger control (6),and temperature control in a mixed tank reactor (7). These investigations have been concerned largely with actual process systems which require a more sophisticated mathematical treatment than that at the disposal of undergraduate students. Therefore, any adaptation of one of these processes for study in a laboratory course leads to many simplifying assumptions which tend to reduce the understanding of and confidence in the results, as well as detract from the significance of the study. A simple controlled process which can be easily operated and easily represented by ordinary linear differential equations with constant coefficients is desirable for such a study. Few simplifying assumptions should be necessary and the parameters must be capable of accurate prediction as well as be easily determinable by experiment. The system should also be easily simulated on linear analog computer equipment.
Figure 1.
Block diagram of process.
Such a system is illustrated by the block diagram in Figure 1and the photograph in Figure 2.
Undergraduate Research Figure
2.
Experimental opparotur.
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39, Number I, January 1962
/ 29
of the controller can be expressed as follows:
Component RepresentationP
The process consists of two tanks with openings cut in one end. These openings are shaped and placed such that water flows from the tank at a rate proportional to the liquid level in the tank. These openings are called Sutro weirs. Design specifications are given by Soncek et al. (8). The tanks are rectangular, 12 in. X 24 in. X 10 in. deep. The dynamic equation rigorously relating liquid level, inflow rate, and time is as follows:
where Qiis the inflow rate, hl is the liquid height in the weir, R is the weir resistance, and T is the product of the tank capacitance and the weir resistance. The tank capacitance is expressed in ft3 per ft of tank height. T has dimensions of time and is known as the time constant or lag coefficient. For the tanks used, T equaled 19 seconds. If the two tanks have identical weirs and dimensions and if the discharge of one is allowed to flow into the other, the equation relating inflow Qi to tank 1 and liquid level ha, the liquid level in weir in tank 2 is as follows:
The water inflow is controlled by a 'lrinch air operated control valve. The valve action can be adequately represented by the first order differential equation:
p is the output signal of the controller: the constants
K,, l / T I , and T Dcorrespond to controller settings for proportional, integral, and rate response, respectively, and e is the difference between the actual tank level and the desired tank level expressed in per cent of the instrument span. The controller receives a measurement simal in the form of an air pressure from a Foxboro "CP" position transmitter. This instrument with the aid of a float converts the tank level to a proportional air signal. There is virtually no lag in the transmitter; consequently its action may be represented by Referring to the block diagram, Figure 1,an additional mathematical relationship is needed : &, the controller setting or desired tank level, and 00, the measured tank level, are expressed as per cent of total instrument span. Equations 2 through 7 are combined using Laplace transform notation as follows:
This is the transfer function of the system The Experiment
where x is the valve stem position and p is the air pressure signal from the controller. T , is the valve time constant. Over small ranges the equation: Qi
=
KG
(4)
can be used to relate flow rate to valve stem position. The air signal for the control valve comes from a Foxboro M53 recorder with an M56-three mode controller integrally mounted. The dynamic action
2
C
Nomenclature used in this representation:
tank capacitance, fts/ft error, %of instrument span level in tank 1, ft h, level in tank 2, ft hs K, = valve gaiin, ft/Yo output signal K. = valvegain, fta/sec/ft & = transmitter gain, %instrument spm/ft K, = proportional gain, Yo output signal/~oinstrumentspan = output signal of controller p Qi = BOWrate through vdveftJ/sec R = weir resistance, sec/ftz 1 = time, see T = lag coefficient of tank, see T. = lae coefficient of valve, see T r = integral time, sec To = derivative time, sec z = valve position, ft s = laplace operator = output signal, Yo span 00 = reference input, O/, spm 0; go(%) = transformed output s~pnitl e,(s) = transformed set point e
30
= = = =
/
Journal of Chemical Edumtion
The following procedure is recommended when using this equipment in unit operations laboratory or in process control laboratory : 1. Determine the lag coefficient T by measuring the tank dimensions and using a plot of discharge rate versus liquid level. T is the product of the tank area and the weir discharge resistance. Determine T also by making a step change in the input flow rate and recording the liquid level variation with time. The time required for the level to achieve 63.2% of its total change is equal to T. Determine K z from a plot of flow rate us. valve stem position. Determine T , experimentally by observing the valve response to a step change in p. Methods for determining a lag coefficient from a response curve are presented by Ceaglske (3). Finally, calculate K , and K a for dimensional homogeneity: These measurements should help give the student a familiaritv with. and an understanding- of. , the conceots of resistance, capacitance, lag, and gain. 2. Simplify equation (8) for the case involving one tank and two modes of control, integral, and proportional. Using the parameters determined in step 1, above, determine the control settings for critical damping, overdamping, and underdamping. Check the results of these calculations by setting them on the controller and observing the response to a step change in set point. 3. 1Make various adjustments in the controller using all three modes of control to demonstrate qualitatively the effects of the various types of control. 4. Using equation? 2 through 7, simulate the entire
control system on an analog computer. Compare the response to step changes in the set point of the simulation to a comparable change on the actual system for various control settings. 5 . Vary the signal representing the control setting on the analog computer simulation with an electronic sine generator. Observe how the amplitude of the output varies with frequency. Ohserve also how the output sine curve lags the input sine curve. Evaluate controllability of the system. All of the above mentioned tests have been run on this equipment. The two methods for determining T mentioned in step 1 produce results which agree perfectly. A curve showing the response to a step change in set point is shown in Figure 3. Such curves obtained by using the procedure described in step 2 agreed very well with the predicted curves for all degrees of damping. Both the closed loop and open loop simulation on analog computer equipment produced response curves in excellent agreement with those obtained from the operation of the process itself.
those that can be carried out with this equipment. Tanks with different lag coefficients may be used. Long pneumatic transmission lines and valves with other characteristics may be used to determine their respective effectson response and controllability. The system may be frequency-tested by using a pneumatic sine generator to pulse the valve. Similar systems using other process media may be designed for faster or slower response. For example, an all pneumatic pressure process would provide a system with a much shorter response time; consequently, it would be much more suitable for the study of frequency analysis techniques. Either of these systems would be relatively inexpensive to construct. Control and measuring instrnments and a control valve would cost $750-$1000; the remaining components can be easily fabricated. The ease of operation, the excellent quantitative checks on calculations, and ease in simulation on a computer make the liquid level process a very powerful device to convince students of the validity of the dynamic equations arising in process control problems. The process gives a realistic demonstration of control in action. Familiarity with this system should enable chemist or chemical engineer to approach a complex process study involving both reaction kinetics and unsteady state behavior with greater confidence; the system will also help familiarize the student with analog simulation techniques. Literature Cited
Figure 3.
Level responses to step changes in controller set point.
I n order to facilitate data collection, a linear sliding arm potentiometer was attached to the float lever arm and the output measured by a Brush Mark I1 recorder. This instrument provided a much faster chart speed for recording the output signal. The device was also useful in determining the lag coefficient of the valve. The transient and frequency testing analyses used are those commonly found in process control texts, such as Eckman (t),Ceaglske (S), and Tucker and Wills (9). Modiflcolions
The above mentioned experiments are only a few of
ZIEGLER, J. G., AND NICHOLS,N. B., ASME Trans., 65, No. 5,433 (1943). D. P., "Process Control," John Wiley & Sons, Inc., ECKMAN, New York, 1958,pp. 269-98. CEAGLSKE. N. H.. "Automatic Process Control for Chemical ~nsneers,';John Wiley & Sons, Inc., New York, 1956. no.98-181. ~-(4) Anonymous, Chem. and Eng. News, 39, No. 6, 11&24 (1961). W. B., ISA Journal, 6, NO.1,45-50(1959). (5) FIELD, (6) HAINSWORTH, B. D., TWY,V. V., AND PAYNTER, H. M., ISA Journal, 4 , No. 5,230-5 (1957). R . W. E., AND PINE,J. F., Conk01 (7) W O R ~ Y C., W., FRANKS, Engr., 4 , No. 6,97-104 (1957). (8) SOUCEK, E., HOWE,H. E., AND MAVIS,F. T., Eny. NewsRecord, 117, pt. 2,679-80 (1936). G. K., AND WILLS,D. M., "A Simplified Technique (9) TUCKER, of Control System Engineering!' Minneapolis-Honeywell Regulator Company, Philadelphia, 1958,pp. 3-76. ~
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