Analysis of Some Common Loops I 83 FIG 3.9. Agitation reduces the effective dead time while increasing the effective time constant. Time sort of response could be represented by dead time plus a single capacity. Thus the simple model just postulated is quite valid, if imperfect. The Analyzer Dynamics associated with the analysis play an important role in the performance of the loop. The foremost limitation in the speed of analysis is generally t’hat of transport’ing the sample to the detector. Fortunately some composition measurements can be made without withdrawing a sample: electrolytic conduckivity, density, and pH are notable examples. But any analysis requiring the withdrawal of a sample, particularly if that sample must undergo a certain amount of preparation, results in a significant accumulat,ion of dead time (see the example cited at the close of Chap. 2). Naturally any effort spent in minimizing the sampling time will be rewarded by bot’h tighter control and faster response. Some analyzers are discont,inuous. They produce only one analysis in a given time interval. This characteristic is worthy of much more attention, because it periodically interrupts t,he control loop. Process chromatographs are the principal, but not sole, constituents of this group. The response of this kind of control loop will be given extensive roverage in Chap. 4, and methods for coping with it will be presented. A few analyzers exhibit a time lag in addition to the dead t’ime asso- ciated with sample t,ransport. Sormally this property is of little conse- quence, except when the process itself consists of nothing but the volume of a pipeline, whose time const,ant may be less than that, of the analyzer. Those measurements which are fast, are by the same t,oken subject to noise. Conductivity and pH are usually in this category, because they are fast enough to react to an incompletely mixed solution, or particles of an immiscible phase. Dead time in sample lines is understandably constant. Dead t’ime in a pipe carrying the main st,ream varies with flow. Dead t,ime within a stirred tank is slight,ly affected by flow, to the extent of F/F,; in most systems t,his variation would not be significant. The natural period of the composition loop would therefore be virtually constant, producing 84 1 Ud n erstanding Feedback Control constant dynamic gain, except for a process whose dominant element is a pipeline. Most analyzers are not so far from being linear that they materially affect the gain of the control loop. The notable exception is, of course, the pH measurement, whose general properties have already been pre- sented. But analyzers are generally given a high order of sensitivity, because of the importance placed on quality control. As a result, the gain of a composition-control loop is invariably high. Objectively, com- position is not as difficult to control as flow, for example, but the specifica- tions placed on product quality are so stringent that ordinary perform- ance is seldom acceptable. The impurity of a product stream leaving a fractionator, for example, may be specified at 1.0 f 0.2 percent. It is virtually impossible to regulate flow within + 1 percent in the unsteady state, yet the composition controller is asked to perform five times as well. This is perhaps the greatest single reason why composition control has the distinction of being a problem area. Because quality can be measured to 0.1 percent is apparently reason enough to expect it to be controlled to the same tolerance. Process Gain The dimensional gain of the process in Fig. 3.8 is the derivative of composition, Z, with respect to concentrate flow X. A material balance on the measured component is simply X = Fx Then, Since the nominal flow F has already been identified as a constant, process gain is also constant. (This is another illustration of the case where process steady-state gain varies with flow, but the time constant does too, so dynamic gain is invariant. Steady-state gain, as calculated above, is only meaningful at the rated flow F.) Dimensional gain of the composition process can always be found by writing a material balance across it. If composition of an effluent stream is controlled by manipulating an influent stream, as in this example, the process is linear. But if effluent composition is controlled by manipulat- ing the efluent flow, the process is hyperbolic: x=1 F dx X dF= F2 (3.28) Analysis of Some Common Loops I 85 (This was already encountered in the temperature-control example where coolant temperature was adjusted by manipulating its flow.) Examples of both linear and hyperbolic processes are common in both composition and temperature applications, because the controlled variable is always a function of the ratio of one varmble to another. If the manipulated variable happens to be in the numerator, the process is linear. example.3.6 The process in Fig. 3.8 is intended to deliver a solution at a nominal flow F, of controlled composition Z, by adding a manipulated flow X of concen- trate to the diluent stream. Let the volume of the vessel be 100 gal and the nominal flow 20 gpm. If mixing is 95 percent complete, then O.O5V/F will be the effective dead time in the vessel: 100 7d = 0.05 - = 0.25 min 20 The balance is a first-order lag: r1 = 0.95 g = 4.75 min Let the sampling time also be 0.25 min, with a 3.0-set analysis lag. The total dead time in the loop is then 7,~ = 0.25 + 0.25 = 0.5 min Without the 3.0-set lag in the analyzer, the natural period would be 4rd = 2.0 min The phase shift of the 3.0-set lag at a period of 2.0 min is & = -tan-1 ?$!!&z = -9” A control valve with a 3.0-set lag will contribute another 9”. This added phase shift extends the natural period to approximately 7,, = 2 o 180 + 9 + 9 = 180 2.2 min The dynamic gain of the process is simply that of the principal time constant: 2.2 G1 = Gl = 2a4.75 = 0.0737 Dimensional process gain is the percent composition change brought about by a change in concentrate flow at the rated throughput: dx 1 -Z-C dX F lOO%PO gpm = 5 %lgpm 86 1 Ud n erstanding Feedback Control Because the process is linear with respect to concentrate flow, a linear valve is chosen. Let the maximum flow of concentrate be 2 gpm. Then G, = 2 gpm/lOO% = 0.02 gpm/% To illustrate the close tolerances to which product quality is generally specified, the analyzer range will be chosen as 4.5 to 5.5 percent, with a normal set point of 5.0 percent. The span is 1.0 percent: GT = 100 %/I % = 100 The proportional band necessary for >i-amplitude damping is finally esti- mated as 200 t’imes the gain product: P = 200(0.0737) (5 %/gpm) (0.02 gpm/ %) (100) = 147 % For a process that is really not very difficult, this is quite a wide propor- tional band. An extremely wide band may then be expected in a truly difficult application, indicating how sensitive composition loops are to changes in load. To summarize, composition loops are principally comprised of dead time plus a single capacity and are noted for high transmitter gain. As a result, a wide proportional band is usually needed, leaving the controlled variable quite susceptible to load changes. CONCLUSIONS The purpose of this chapter has, been to acquaint the reader with the properties of common process control loops and the reasons for these properties. Analysis served as a useful tool to present the case, while TABLE 3.3 Properties of Common Loops Property Flow and liquid pressure Gas pressure Dead time. No No Capacity. Multiple Single Period. l-10 set Zero Linearity. Square Linear G,GT 2-5 0.5-l Integrating Noise. Always None _____ Proportional 100-500 % O-5 % 50-200 ‘f!$, Reset. Essential Unnecessary Derivative No Unnecessary Valve. Linear Eq. percent * Applies to liquid pressure. Liquid level No Single l-10 set Linear Integrating Always 5-50 70 10-100 % Seldom Yes No Essential Linear Eq. percent Tempera- ture and vapor pressure Variable 3-6 Min - hrs Nonlinear l-2 None Com- position Constant l-100 Min - hrs Either lo-1,000 Often ___-__ loo-l,OOO% Essential If possible Linear Analysis of Some Common Loops I 87 at the same time demonstrating how to identify the significant elements in a loop. Rarely will a flow or level loop need analysis, but when com- position-control problems arise, this procedure can be of inestimable value. Much of what has been derived and weighed and discussed in the fore- going pages is summarized in Table 3.3. Nothing that has not been already covered is presented in the table, yet gathering all this information together discloses some interesting features. Notice, for example, the similarity between level and flow loops, with respect to both natural period and the presence of noise. Without any doubt, however, each of the five groups above is separate and distinct from the rest. REFERENCES 1. Bradner, M.: Pneumatic Transmission Lag, ISA Paper No. 48-4-2. 2. Catheron, A. R.: Factors in Precise Control of Liquid Flow, ISA Paper No. 50-B-2. 3. Considine, D. M.: “Process Instruments and Controls Handbook,” chap. 2, McGraw-Hill Book Company, New York, 1957. 4. Esterson, G. L., and R. E. Hamilton: Dynamic Response of a Continuous Stirred Tank, presented at the Joint Automatic Control Conf., Palo Alto, Calif., June, 1964. PROBLEMS 3.1 -1 volume booster installed at the inlet to the valve motor of Example 3.2 reduces its time constant to 0.5 sec. Predict the period of oscillation that Iv-ill result from the change, allowin g- 45” phase lag in the proportional-plus-reset controller. Calculate the proportional band and reset time for jb-amplitude damping. 3.2 What would be the estimate of the natural period and proportional band in Prob. 3.1 if the dynamic elements were all assumed to be dead time rather than capacities? Is this a valid approximation? Why? 3.3 Let pressure downst’ream of the valve in Example 3.2 be controlled instead of flow. -kt no flow, there is a static head of 5 l)sig, while 10 gpm will raise the pressure to 13 psig; the range of the pressure transmitter is 0 to 25 psip. Estimate what the proportional band of the controller will be for ;/,-amplitude damping with the period and reset time used in the example. 3.4 A mercury manometer capable of reading f 15-in. differential pressure is used to indicate the flow in a gas stream. What is its natural period? H OW would it affect the control of flow? 3.5 TO verify the choice of an equal-percentage valve for Example 3.5, cal- culate the process gain and the product of process and valve gain for heat loads of 5000, 10000, and 15000 Btu/min; assume that. the difference between con- trolled reactor temperature and average coolant temperature varies linearl) with heat transfer rate. 88 1 Ud n erstanding Feedback Control 3.6 Two fluids are blended in a pipeline 20 ft upstream of where the mixture is sampled. The pipe contains 0.4 gal/ft of length, and the flow rate of the blend varies from 10 to 80 gpm. Dead time in the sample line to the analyzer is 15 sec. A circulating pump is installed to maintain 100 gpm flow through that 20-ft section of pipe without affecting the throughput. Compare the natural period for integral control with and without the pump in operation. What else does the pump provide? 3.7 In the same process, the flow of additive is manipulated through a linear valve whose maximum flow is 1.2 gpm. The range of the analyzer is 0 to 1 percent, additive concentration. Estimate the proportional band required for at least ~-amplitude damping if the reset time is set for 60’ phase lag with the pump operating. [...]... 0.517&1 4. 00 0. 64 5.37 0.86 - - - - 0.9772/71 0.81 0.96 125P 100 125 2.867d 4. 00 6.79 2 05rd2/r1 1.62 2.92 2.867d 1.66 1.27 0.51Td 0.88Td 0. 64 0. 64 0.86 0 .49 _- - - Interacting 0,907d three-mode ? 1 0. 64 ,0. 54 0.9OTd 0. 64 0. 54 PR/100 70 m 0 0 0 Gain Performance, P _ Proportional plus reset Process 1 04 115 141 I - -15 -30 -45 +36 22%,/T, 0 2 54 - 3 6 540 0.64rd/r, 0 .45 rJn 0. 64 1.08 m Linear FIG 4. 9 Increasing... extension to a broad range of difficult processes, a dead-time plus integrating process will be used, with controller settings left in terms of 74 a n d 71 This is necessary because the natural period of a loop does vary with the controller settings Table 4. 2 has been prepared by equating the gain product of process and controller to O.Tj, with gain and phase for the controllers accurately calculated from... (TV = 27d) 3 Availability of critical damping FIG 4. 12 The amount of damping varies inversely with loop gain 0 1 2 3 4 n=t/r* 5 6 7 0 Linear Controllers FIG 4. 13 A proportional-plus-reset controller is the complement of a single-capacity process Integral control of dead time was able to eliminate offset, but at 7O = 4rd Neither proportional nor integral control was capable of critically damping the loop... tangent of 22.5’ is 0 .41 4 Reset time is then 2.3rd R = 2n(0 .41 4) = 0.887d Quarter-amplitude damping requires loop gain to be 0.5 GPRKp = 0.5 = 100Kp4 1 + (0 .41 4)2 P P = 2OOK,4i3 = 216K, Integrated error is then E E am = 100 = (2.16KP)(0.887d) = 1.90KPTd This indicates that complementary feedback significantly reduces IAE, as well as settling time It is important to see over what range of processes complementary... complementary controller on simple processes, it is nevertheless interesting to speculate on its configuration If the process is a first-order lag, its complementary controller turns out to be proportional-plus-reset In fact, pneumatic two-mode controllers are made this way, as shown in Fig 4. 13 A single-capacity process can tolerate zero proportional band and zero reset time Compared to a dead-time process, ... procedure for three-mode controllers: 1 With maximum reset time and minimum derivative, excite the closed loop into oscillation by reducing the proportional band TABLE 42 2 Determination of Optimum Settings for Two- and Three-mode Controllers ,Controller - Modes D Gain _ ” deg _ _ 1OOP R Noninteracting three-mod< 0 0 127r&, 4. OOsd 158 220 361 4. 80 0.76 0.95 6.00 8.00 1.27 - 4 52sd2/rl 3.65 4. 58 109P +23 llOTd/T,... less difficult processes, lower damping will enhance recovery from load disturbances due to greater controller gain In general, better performance will be obtained on more difficult applications by using delayed reset ,4 as shown in Fig 4. 17 This is obviously a compromise between two-mode control and complementary feedback, 3 4 110 1 Selecting the Feedback Controller *&pz* difficult FIG 4. 17 Three lags... from the initiation of the step The actual track of the controlled variable c is shown as a broken line, while its sampled value c* is indicated by the solid line Linear Controllers c,, 2 FIG 4. 18 The dynamic characteristics of the slow process are less affected by sampling // c* Fast I 111 process - c I fi I 5[r -4 0 1 2 3 4 ” = t/Al 5 6 7 If a process can respond fast enough, its dynamic characteristics... especially useful for control, unless it is “held” or memorized until the start of the next interval A series of steps (Fig 4. 18) is thereby generated as the process changes state Additive FIG 4. 19 Flow blending processes are often dominated by a discontinuous analyzer Main stream - Blend - 0 11 P 1 Selecting the Feedback Controller Although feedback control can be exercised over a process containing... two-mode controller will be applied to the same process, but it must be adjusted so that the error will not cross zero during recovery Thus integrated error is actually IAE, permitting comparison of loops with different damping A phase % 5 FIG 4. 14 A load-induced error prevails for one dead time % h ‘7 AmKp c r l -4 -A Time 108 1 Selecting the Feedback Controller lag of 22.5” is chosen for the controller, . 0 125 -36 P 127r&, 158 220 361 llOTd/T, 127 196 22%,/T, 2 54 540 Process 70 Gain 4. OOsd 0.64rd/r, 4. 80 0.76 6.00 0.95 8.00 1.27 -__ 3.197d 0.517&1 4. 00 0. 64 5.37 0.86 2.867d 0 .45 rJn 4. 00 0. 64 6.79 1.08 Perform- ance, PR/100 m 4 .52sd2/rl 3.65 4. 58 0.9772/71 0.81 0.96 2 0.86 Interacting three-mode _- ? 1 , D 0 0 0 0 __- 0,907d 0. 64 0. 54 ,Controller R m 2.867d 1.66 1.27 0.88Td 0. 64 0 .49 0.9OTd 0. 64 0. 54 - I - Gain ” deg ____ 1OOP 0 1 04 -15 115 -30 141 -45 109P +23 100 0 109 -23 __ 125P . engineers have prepared 1+D& t -4 y(lt$/dt) m I- (a) (b) FIG 4. 8. Three-mode controllers: (a) interacting, (b) noninteracting. Linear Controllers I 101 TABLE 4. 1 Effective Values of Modes at