Chapter 18. Process Control

Transcription

1 Chapter 18. Process Control 18. INTRODUCTION The atheatical odels described in previous chapters serve adequately for successful configuration and ipleentation of unit processes as well as distributed and coplex processes. The basic preise in the applications of these odels has been that the operation was under steady state conditions. In actual practice however, the steady state condition is easily disturbed by changes in operation variables when the atheatical odels are affected. If an ideal steady state condition was taken as the ean of the variations and the deviation fro the ean deterined, then with proper instruentation it ay be possible to return to steady state operation. Over the years, suitable instruents have been devised for the purpose. They are activated by electric signals generated by individual variables. These signals are prograed to recognize the deviation fro the ean. They then suitably operate to restore noral conditions. The ain objective for process control therefore is to establish a dynaic atheatical odel, onitor the deviation fro the odel and finally restore the original conditions of operation. The process of controlling a dynaic syste is coplicated especially in ineral processing systes where a nuber of variables are involved siultaneously. Developents towards autoatic control of plant operations have been coensurate with the developent of coputer science and instruent technology. Its ipleentation has resulted in consistent plant perforance with iproved yield and grade of the product with less anpower. The ter "process control" therefore refers to an engineering practice that is directed to the collection of devices and equipent to control processes and systes. Coputers find application in siple systes, such as single loop controllers and also in large systes as the Direct Digital Controller (DDC), Supervisory control systes, Hybrid Control Systes and Supervisory Control and Data Acquisition (SCADA) systes. Further developents in process control are supported by any secondary concepts such as coputer aided Engineering (CAE). The coonly used present syste is the Distributed Control Syste (DCS). It is ade up of three ain coponents, the data highway, the operator station and the icroprocessor based controllers. The data highway handles inforation flow between coponents ensuring effective counication. The icroprocessor controllers are responsible for effective control of the processes and are configured to handle as single or ultiloop controllers. The operator station allows the control coand to be given, aintain the syste data base and display the process inforation. The displays norally used are the group and detail displays, trend displays and alar annunciated displays. In this chapter we will priarily discuss how atheatical odels with appropriate instruentation are used for controlling the quality and quantity of yield of a ineral processing operation.

2 Controller Modes Process control systes can be divided into two ajor groups: 1. Continuous control that involve onitoring and controlling of events continuously, 2 Digital controls that involve the use of coputers and icroprocessors. In this book the continuous control syste is dealt with ore as it fors the basis of the present digital syste. Controlling, say the level of a flotation tank which is being filled continuously and fro which the pulp is withdrawn continuously, can be done crudely by observing the rise (or fall) of level and restoring it anually by anipulating valves and increasing or decreasing the input flow rate to the tank. Such an onoff ethod would result in an unsteady profile of level (Fig. 18.1). This situation is unacceptable in ost ineral processing circuits. To solve the proble instruents have been devised and strategies developed to iniize the fluctuations in level. Autoatic controllers have therefore being devised which serve to control flow rates, density of slurries, tank and bin levels, pup operations and alost all unit operations like crushers, ills, screens, classifiers, thickeners, flotation vessels and aterial handling systes. The two basic control strategies or odes of these controllers are known as: 1. feed back control syste, and 2. feed forward control syste. In the feed back control syste the output fro a process is onitored continuously by a sensor. When the output changes the sensor detects the change and sends signals to a coparator which copares the signal with the set point for noral steady state operation. It then estiates the error or the deviation fro the ean. The error signal is passed on to the controller which copares the signal with the true set point and sends a signal to an operating device to reduce the error to zero. The signals are electrical, echanical or pneuatic devices. Fig is a typical block diagra illustrating the feed back syste Signal Mean level Tie Fig Manual onoff control of flotation cell level.

3 624 Final controller > Qp t Feed Set point ", e ^ Controller ^ Process. Output Coparator Sensor Fig Block diagra of Feed Back control syste. It can be seen that the coparator has three functions. Its first function is to correctly receive signals of easured value fro the signal onitor. Its second function is to copare the signal with the set point and then copute the deviation against the nor (set point) and its third function is to activate the final controller to correct the error. There are two process factors that ake the feed back control unsatisfactory. These are the occurrence of frequent disturbances, often of large agnitude, and the lag tie within the process between occurrence of an event and delay in recognizing the signal. As shown later, these disturbances and lag ties can be easured and corrective steps applied. In the feed forward set up, the input signal, say of the feed, is onitored and controlled prior to the feed entering the process. In so doing it is expected that the feed to the process is unaltered and therefore the process perforance reains unaffected. A block diagra of the feed forward syste (Fig. 18.3) illustrates the principle of its operation. In this set up the indicator in the input strea indicates the deviation in the input strea characteristics, (like feed flow rate) to the controller. The controller confines its activity to the incoing strea (and not on the process), coputes the agnitude of the error and signals to the controller to provide appropriate action to restore the input strea characteristics to its original level. Input Control eleent t Servoechanis t Measuring device T Controller Process r Indicator Dutnut Fig Block diagra of Feed Forward syste of control.

4 625 Controllers are designed so that the output signal is: 1. proportional to the error 2. proportional to the integral of the error, 3. proportional to the derivative of the error, 4. proportional to a cobination of the odes. When the output signal O, is proportional to the error e, it is known as proportional controller. Matheatically the control action is expressed as: O = G P e (18.1) where Gp is the proportionality constant and e the error. Gp is usually known as the gain. It can be seen that the gain is the ratio of the fractional change in the ratio of the output to input signals. When e = 0, the output signal is also equal to 0. That is, no signal is eitted fro the onitor. In this situation Eq. (18.1) is written as: O = O O + G P e (18.2) The proportional operation is expressed as proportional band. The band width is the error to cause 100% change on the etering gauge or chart. The integral controllers are known as reset controllers. They are so designed that the output is proportional to the tie integral of the error. Thus the output signal, O, is given by: O = Gi \e.dt (18.3) o where Gi is a constant. For integral ode the reset action is ore gradual than the proportional controllers. The derivative ode of controllers stabilizes a process and the controller occupies an interediate position. An exaple would be the onitoring of bubbling fluid level where only the average fluid level is easured and onitored, like the level in a flotation cell. The output signal in the derivative ode is expressed as: O = G D^ (18.4) at where G D is the constant. In practice, the proportional ode is usually cobined with integral or derivative odes but ost of the tie all the three odes are cobined. In each cobination the output is an additive function, that is for: 1. For proportional and integral (P+I) ode: O = O o +G P e+gi e.<# (18.5)

5 626 For proportional +derivative (P+D) ode: 'de" = O 0 +G P e +G D dt (18.6) 3. For proportional +integral +derivative ( P+I+D) ode: O = O o + G P e + Gi f e.dt + G D ( * {dt (18.7) When any controller receives a signal fro a sensor, the response tie depends on the ode of the controller. Of the three odes, the response of the P+I+D (PID) controllers is the fastest. The P+I (PI) controllers takes slightly ore tie, while the P+D (PD) and P controllers never return to the original situation but reains at a level. The difference between the original level and the new steady level of proportional controllers is known as offset, or the droop. The offset value is therefore the difference between the steady state and the required control level or set point. In the P+I or P+I+D control systes no offset is necessary Fig illustrate the relative control tie taken by controllers operating on different odes. In the operation of a P+I+D controller the derivative ter signifies the rate of control action on a process affected by a disturbance Signals and Responses Any signal resulting fro a disturbance during an operation is known as a forcing function. The coon types of forcing functions are: 1. step function, 2. pulse function, 3. decaying exponential function, and 4. sinusoidal function. 5 4 Deflection P+I Tie,secs. Tie.secs P P+D P+I+D Off set Fig Responses by controllers subjected to unit step disturbance.

6 627 The step function is instantaneous It involves no tie as illustrated in Fig agnitude of the signal is given by the height X and u(t) is unit step at tie, t = 0. condition can be atheatically expressed as: The The (18.8) When a disturbance is repeated at regular intervals it is described as a pulse function. It ay be described as two equal functions of agnitude X operating in opposite directions. Fig illustrates a pulse function, where the pulse duration 0 1 is repeated. As can be seen in Fig. 18.6, the pulse function is tie based and ay be defined as: fo, t<0 /(t)= x, 0<t<t 0 (18.9) where x = the height of the function and t the width. ƒ(t) /(t) X Tie,t. Fig Step Function. ƒ(t) X J L 0 1 Tie, t Fig Pulse Function.

7 628 The ipulse function is a special case of a pulse function in which x = k/to, where k is a constant. It eans that the area (x to) is a constant. A decaying exponential function shown in Fig ay be written as: /(t) = x(t)e" st (18.10) Here: /(t) = 0 at t < 1 and equal to e" st at t > 0 where x(t) is the unit step. A sinusoidal tie function is described by the equation: /(t) = sincot (18.11) f (t) Tie, t Fig Exponentially decaying curve f (t) N 2.2 ID IO IO <D <D tie,t Fig Sinusoidal function.

8 629 and its properties are: f 0, t < 0, /(t)h (18.12) [xsincot, t>0 A sinusoidal tie function is illustrated in Fig Forcing functions are generally represented by first or second order differential equations. That is: A o y =X(t) Zero Order (18.13) i^i + Aoy =X(t) First order (18.14) dx ^ll. + Ai ^ + A o y = X(t) Second order (18.15) dx dx where X = input variable or input function and Y = output variable, A o, Ai and A 2 are the constants. For convenience of atheatical anipulations in control systes, Eqs. (18.13) (18.15) are written in the for of Laplace transfors. On integration the tie doain is obviously eliinated and a duy operator is introduced. Thus, sequentially, the three equations transfors to: Y(s) 1 = l (1816) Y(s) X(s) 1 (ra + 1) (18.17) Y(s) X(s) 1 T 1 S + T 1 SX 2 S + T 2 S 1 (T IS + 1)(T 2S S + 1 (18.18) where s is the Laplace duy operator and T the tie constant. The advantage of using such a technique is that the transfors can be added and subtracted by siple algebraic rules. After the operation the result ay be inverted back to tie doain (like taking logs and antilogs of nubers). Derivation of the equations are explained in Appendix C Input and Output Signals of Controllers The relation between the output and input signals of controllers is conveniently expressed by the ratio of the output and inputs transfors. The ratio of the transfors is known as the transfer function. In this book this is referred as TF. The transfer functions of different odes of control systes can be easily deterined. Where the error is e and Gp, the gain and T the tie, the transfer functions of different control odes are given in Table 18.1.

9 630 Table 18.1 Transfer function of controllers Controller Transfer function Proportional (P) Gp x Proportional+Integral (P+I) G P = [(T i s +1 / x i s ] Proportional+Differential (P+D) G P = [ 1+x D s ] Proportional+Integral+Differential G p [ (iis +TS*T2S +2) / TI s ] (PIP) The use of Laplace transfors to deterine the response of a function of first and second order differential equations are illustrated in Exaples 18.1 and Exaple 18.1 Let us assue that it is required to solve the following first order differential equation by the use of Laplace transfors. dx + 2x = l (18.19) dt Based on the condition x(0) = 0 Solution Step 1 Using Laplace transfor table in Appendix Cl, Eq. (18.19) transfors to: sx(s) +2x(s) = or x(s) [ s+2] = (18.20) s s Using the theory of partial fractions, Eq. (18.20) ay be split as: x(s)= = + (18.21) s(s + 2) s (s + 2) where A and B are constants. Step 2 The proble now resolves to deterining A and B. To deterine A, ultiply both sides of Eq. (18.3) by s to give: 1 A ^ (1822) (s + 2) (s + 2) Eq. (18.22) holds for all values of s. Setting s = 0, A = V 2 To deterine B, ultiply both sides of Eq. (18.21) by (s+2) giving:

10 631 s s Eq. (18.23) is true for all values of s, Setting s = 2, B = V4. Thus Eq. (18.21) converts to: 2s 2 This inverted to tie doain gives: x(t) = 0.5e" 5t (18.25) Exaple 1S.2 Deterine the response to a unit step function applied to a process defined by the second order differential equation: Initial condition is x=0 Solution Stepl Putting ^ = s, y(t)) =Y(s), and /(t) = F(s) ax we get, (s 2 +5s+6)Y(s) = (s+l)f(s) (18.27) Since a unit function is involved F(s) = (Appendix Cl) s ThatisY( S ) = fr* 1 ) (18.28) ( + 2)( + 3) Expanding Eq. (18.28) by the partial fraction ethod: (18 29) For estiating A, take Lt s * 0, next putting s = 0 as in Exaple 18.1 and substituting in Eq. (18.29): A = L t ( s + 1 ) 1 1 (1830) s(s + 2)(s + 3) 2x3 6

11 632 Siilarly for estiating B take Lt s > 2, next putting s = 2 in Eq. (18.29): (s + 1) B=Lt_ s^2s(s 2 4 ^ =., = (18.31) V + 3) 2(2 + 3) 2 ' And siilarly for estiating C, take Lt s > 3, next putting s = 3 in (18.29) C=Lt fr±*u 2 + \=l (18.32) ~ 3 V ; s(s + 2) 3(3 + 2) 3 Thus Eq. (18.29) can now be written as: Y(s)= + x^ x= (18.33) 6s U s + 2j U s + 3j Eq. (18.33) can now be inverted to tie doain, t, using table in Appendix C2 getting: Y(t) = + e" 2t e" 3t (18.34) K ' Integration of Processes and Block Diagra Laplace transfors and the transfer functions are conveniently used to study the input and output of processes in series or parallel. For exaple, for two identical unit processes operating in series, if the transfer function of each of the process is T(p) then the input and output of each process can be illustrated in Fig and Fig According to definition, the transfor T(p) is given by: T(P)=y (18.35) and the transfer function for process 2 shown in Fig will be: T(p)=~ (18.36) If the processes were cobined in series then the transfor of the inputs would be the su of Ii +I 2, and the transfor of the cobined output will be: O, + O 2 = T(p) I,+T (p)i 2 = T(p) [Ii +1 2 ] (18.37) If however, three processes were in series (Fig ) and the transfer function of each were Ti(p), T 2 (p) and Ts(p), for inputs Ii, I 2,13, then for each process the transfors will be:

12 633 Fig Block diagra for process 1. Fig Block diagra for process 2. Ti(P) T 2 (P) T 3 (P) Fig Processes operating in series. Ti(p) = f, T 2 (p) = i. ^ T 3 (p) = f (18.38) According to definition the transfer function T(p), for the three processes cobined in series would be: It can be seen that the product Ti(p) T 2 (p) T 3 (p) =.. = * I 4 I 2 I 3 I, (18.39) Thus Eq. (18.39) shows that the overall transfor of a process consisting of a nuber of processes in series is a product of transfors of individual processes. However, when the processes are operating in parallel, as in Fig with a coon input I and output O4 and if the individual outputs were Oi, O2, and O 3 respectively, then again by definition, the transfors for individual unit process will be: Again according to definition the cobined transfer function will be = It can be seen that: (18.40) l[t l (18.41)

13 634 Fig Processes operating in parallel. Thus for processes in parallel the overall transfor can be obtained by adding the individual transfors. With the help of Laplace transfors it is possible to estiate the output transfors of processes in series or parallel Setting and Tuning Controls The responses of controllers are not iediate but take a finite tie to attain the final value. A rule of thub for setting is to study the response curve and to arbitrarily (trial and error) set the tuner such that the decay ratio is 4:1 between the first two successive peaks. The decay ratio being defined as the ratio of the heights of succeeding peaks in the decay curve. However, for setting and tuning two well established ethods have been practiced, naely Ziegler and Nichol [1] ethod and Cohen and Coon [2] ethod. Both ethods are epirical. Both ethods yield siilar results. In the following therefore the ethod by Cohen and Coon only is described. In Cohen and Coon's approach, a feedback syste is set up and the loop is opened. A sall step function (load) is injected. The response is recorded against tie. A block diagra of the set up is illustrated in Fig and a typical response curve is shown in Fig The ultiate response, UR is the steady state value and is given by the asyptote to the curve (Fig ). In order to deterine the paraeters for tuning a tangent is drawn through the point of inflection and its intersection with the xaxis (Dn in Fig ) deterined. This is the "apparent" dead tie. The slope SL of the tangent is [UR/ ta]. where ta is the apparent tie constant. That is: s, = t (18.42) The steady state gain Gc between pulse P and ultiate response UR is: U R = PG C (18.43) Cohen and Coon assued that an openloop syste of a process behaves as a first order syste described by the first order transfer function:

14 635 Opened Loop Loac Controller Valve Chart Recorder r Process Sensor Fig Block diagra indicating Cohen and Coon's set up [3]. 6O _.. τ% B _ τxa A U R 5 I Response n ** < D T r / \ / A / >f > Tie,, t,secs P* c Fig Response curve to unit step change in Cohen and Coon's setup. T (z) p() (18.44) Using the data fro Fig , and fro Eqs. (18.42) and (18.43) Cohen and Coon [2] epirically established rules for deterining the settings for different odes of control.

15 636 The epirical ethod designed by Cohen and Coon (and also Ziegler and Nichols), are being increasingly replaced by the "odel ethods" of tuning [4,5,7]. Two such odel ethods presently used are: 1. ITAE (Integral Tie weight absolute error), 2. IMC (Internal odel control). The ITAE ethod arises fro the fact that the decay ratio of 4:1 for the first two peaks is not necessarily the sae for the subsequent peaks. Thus an error is introduced. To iniizing this error, the process is considered as first order with a tie delay. The outputinput ratio is given by the transfer function: O^ =Gp^ x P s + l I s (ig45) where Os = Output of process, I s = Input to process, Gp = Process gain, XD = Process tie delay, xp xc Is = Process tie constant = Closed loop tie constant = Input to process (Controller output) The controller settings for P+I and P+I+D controllers as derived by the ITAE ethod are given in Tables 18.2 and The noenclature used in the tables refer to Fig The ratio of xa / TB =TR. is taken and the closed loop tie constant is taken as xc The ITAE ethod iniizes the integral of the tie weighted absolute error. The IMC ethod uses a closed loop with a tie constant xc and for: >0 and that x c > 0.1 x A The settings of P+1 and P+I+D controllers for the IMC ethod is also included in Tables 18.2 and Table 18.2 PI Controller Settings [6]. Setting G c T, ITAE i6 G P * R x A [x R ] IMC G P (T C + x B )

16 637 Table 18.3 PID Controller Settings [6]. Setting G c Ti XD ITAE G P [TR1 T A (x R ) x A x X[T R ] 0929 IMC ( 2T^+T B ) >(2r c +r B ) + 2 T^ CATB 2TA+ T B In a particular situation the choice of selecting one or other ethod of setting depends on the acceptance of the nature of the oscillatory response which ost suits the situation. In practice, preliinary controller settings are set according to recoendation given in Tables 18.2 and 18.3, but the final fine tuning is done visually. However ore sophisticated nuerical ethods of quantifying controller responses have been introduced like IAE (Integral of Absolute Error) and ISE (Integral of Squared Error). Flintoff [8] has entioned the use of software. Software like MATLAB is increasingly being used Coplex Advanced Controllers With the increasing coplexities of processes, additional control set ups have been introduced. The basic principles of operation of these coplex controllers are the sae as those already entioned. Of these, three are of direct use to ineral processors: 1. Error squared controllers, 2. Ratio Controllers, 3. Cascade controllers Error squared controller Soe process systes require little control action when the process variable is very near the set point. In such cases, control action can be achieved satisfactorily by using the error squared in the PI and PID algorith instead of siple error ters. Controllers using this technique perit the use of low gain setting when the error is sall and does not hold the controlled variable at a particular set point but operate as an 'average control'. When the disturbance is large the gain is increased to avoid large deviations. According to Gault [9] a "dead band" effect is produced. The dead band effect around the set point can be reoved by using (s+s 2 ) ter as the error. It should be noted that e 2 = e* s which retains the sign of the error Ratio Controller When two loads in a process have to be controlled siultaneously such that the ratio between the reains constant, the controllers used are ratio controllers. As an exaple, say in grinding ill circuits where the feed is set at a definite ore/water ratio, the water flow controller set point has to be set at a definite ratio with the tonnage rate to ensure the pulp density to the ill reains constant. The functional odule that ipleents the ratio, is essentially a ultiplier, thus if P V i and Pv2 are two process variables then:

17 638 R Pv 2 or R = L (18.46) R is known as the settable ratio. The ratio station has siilar features to the PID syste [10]. The prie role of the ratio station is to provide a way of generating the set point. The ratio itself becoes a set point controlled by higher level control in a structured syste. The ratio controllers can be used both in open and closed loops. Such controllers are often used in practice in flotation circuits where the water additions is related to the target feed rate and in circuits where ph control is required, say lie additions, in gold cyanidation process. 18.6,3. Cascade Controller By using a single controller it is difficult to control a process that involves say, two or ore stages of different dynaic characteristics. The proble is aggravated when the response tie of the first or ain controller is longer than the second. In such cases the introduction of a second controller is necessary. The second controller operates in conjunction with the first controller but only at a particular stage of the process where the disturbance can be detected with iniu dead tie and short response tie. Such a control syste is known as Cascade Control. It can be described as a ultiloop control syste. A typical exaple is illustrated in Fig where the objective is to aintain a constant level of a tank. Fig shows that the ain loop controls the tank level. The loop operates slowly copared to the fluctuations of the feed strea. The feed strea in its part is controlled by a faster inner loop that controls the flow. The second controller has an external set point which changes with the level controller and operates at a uch slower rate. Cascading slightly iproves the copensation of disturbances entering the outer loop, but the copensation of the inner loop is greatly iproved. The set point of the inner flow Flow transitter Feed Faster feed control loop Flow controller External set point Main loop slow Level controller (Internal set point) Fig Cascade Controller syste.

18 639 controller is obtained fro the output of the level controller. The cascade control syste provides a high degree of stabilization to the overall control Adaptive Controller An adaptive control syste autoatically copensates for variations in syste dynaics by adjusting the controller characteristics so that the overall syste perforance reains the sae, or rather aintained at optiu level. This control syste takes into account any degradation in plant perforance with tie. The adaptive control syste includes eleents to easure (or estiate) the process dynaics and other eleents to alter the controller characteristics accordingly. The controller adjusts the controller characteristics in a anner to aintain the overall syste perforance. The basic essentials of the adaptive syste are: 1. Identification of syste dynaics, 2. Decision, and 3. Modification. Once the syste is identified, (which is a difficult process) the decision function operates. This in turn activates the odification function to alter the particular process paraeter and to aintain optiu perforance. The coon ways of evaluating perforance are by odel coparison and perforance criteria. In the odel coparison syste, a odel is selected that bears reseblance to the desired syste characteristics. In it, all effects of the syste characters and effects of disturbances are known. According to Gault [9] the response characteristics of the control syste variable paraeters are "slaved" to the response characteristics of the reference odel. The error between the odel and the control syste is acted on. This action is accoplished by an adaptive operation which produces the required syste gains. The path of adaptation is the iniization of the integral of the error square. In the perforance criterion, a general perforance index such as the integral of the error squared is chosen and continuously coputed. The syste is adjusted to keep the value of the index at a iniu level. A developent of this is the Kalin Filter [7]. This is a coplicated process. Interested readers are referred to the work of Stephanopoulos [11], Flintoff and Mular [12], Dead Tie Copensation So far we have assued that a variation in the feed and process will be onitored iediately and corrective action to any disturbance taken iediately. But in practice a tie is elapsed between the change and the tie at which the change is detected. For exaple, an apron feeder discharging ore is set to feed a travelling belt conveyor. A short section of the conveyor is ounted on load cells which easure the ass rate of ore travelling. Tie elapses before any change in the feed ass rate occurs on the feeder and the tie it is detected by the load cells of the conveyor as the load cells are located at a distance (Fig ). This is the dead tie of the controller loop and is often referred to as transportationlag or distancevelocitylag. To design a control loop therefore the transfor for the dead tie has to be considered. Suppose a plant is constructed by soe eans so that the plant transfer function Gs is the transfer function without the dead tie and let TD(S) is the dead tie. A typical feed back block diagra of such a plant would be as illustrated by Fig. 18,17. In this typical setup, unfortunately, the signals cannot be easured directly as the deadtie occurs as a distinct

19 640 eleent. Hence a dead tie copensation devise is sought. Copensation is introduced by taking the output signal and passing it through a odel to cancel the original signal. The ethod is further explained below with an exaple. Suppose a odel is constructed by soe eans so that it is coposed of a dead tie TM.. In a typical feed back syste (Fig ) if an output signal fro the controller is passed through the odel then it can be easured, as seen in Fig In Fig the cancelling odel is added to the noral feed back odel. This arrangeent is known as Sith predictor or dead tie copensator. A successful use of this concept was ade by Anderson et al [13] while designing four SAG ill feeder controls in Freeport, Indonesia Instruentation and Hardware Instruentation The function of the instruents is to easure the steady state and transient behaviours of processes in a physical world. Usually the design of instruents involves ultidisciplinary activities all of which a ineral processor need not be involved, but he should be aware of the reliability, including correct operation of the instruent and also their sensitivities. Controller WJApron feeder Tie LagBelt conveyor Load cells Sensor Fig Dead Tie Lag. Set Point rw Controller G(s) Output Fig Typical Feed Back loop.

20 641 Process Set Point Output Fig Dead Tie Copensator to feed back loop. Further, it is necessary to ensure that their loading effects are inial and that the signal is passed without attenuation, loss of agnitude or phase change. For ost instruents such operation software is available fro instruent anufacturers. Instruental errors are inherent within an instruent arising fro its echanical structure and electronic coponents. This iportant liitation ust be recognized when selecting and using instruents. Most loops are operated by PID controllers which are ipleented by DCS (distributed control systes) operating software. This eans that the control syste is not centralised but is coposed of interconnected ultiple units, hi ost other cases, ratio or cascade loops are eployed. According to Flintoff and Mular [10] an average grinding ill has about 2000 loops of which 97% use PID. All systes operate on 420 A output signals. Multi transitter signal input is available so that different control loops can be rapidly configured. Microprocessors provide a range of control functions. The present day instruentation takes care of the DCS syste and therefore the control echanis is coplicated and is ore in the doain of the instruent engineer. However, the etallurgist and process engineers needs to know the basics in order to best utilize control echaniss. Instruentation and control of ineral processing operations is usually visualized as involving connected building blocks. Process control and instruentation consider these blocks divided into three or four levels: These are: Basic block Supervision block High level block "Watch dog" Level 1 Level 2 Level 3 Level 4 A general description and function of these levels follow.

21 642 Level 1 control This is the regulatory level where basic controls loops like, P+I control loops include control of feed tonnages fro bins, conveyors, anipulating of bins, water addition loop (in illing circuit) pup speed and sup level controls, thickener overflow density control etc are involved (depending on the process circuit). Level 2 This is a supervisory control stage that includes process stabilization and optiizing, usually using cascade loop and ratio loops. For exaple, in a ball ill circuit the ratio loop controls the ball ill water while the cascade loop controls the particle size of product by anipulating the tonnage set point. Level 3 Controls at this level include axiizing circuit throughput, liiting circulating load (where applicable). Level 4 This is a higher degree of supervisory controls of various operations including plant shut downs for aintenance or eergency. Austin [14] has referred to level 4 controls as "watchdog" control. The usual instruents covering the regulatory levels in ineral processing plants are shown in Table 18.4: Table 18.4 Instruents and their uses in ineral processing plants Instruent Property easured and use Weightoeter Measures ass flow rate of aterial. Usually a sall length of conveyor belt rests on load cells Level indicator Measures level by, ultrasonic, pressure differential or siply by a floatball Flow eters Measures ass rate of flow generally by noninvasive ethods, like Magnetic flow eters, Ultrasonic flow eters.(doppler effect) The invasive types include orifice flow eters. Density gauges Measures fluid and pulp densities either on strea with Gaa ray density gauges, or by taking saples and using Marcy density gauge. Pressure gauges / Measures pressure or pressure differentials pneuatic,/hydraulic Therocouples systes and teperatures Particle size Measures particle size passing a particular sieve size, e.g., 75 analyzer icrons on strea (OSA) by generating ultrasonic signals ph indicator D/AA/D converter Attenuators For easuring acidity and alkalinity of solutions For converting digital to analog signals and analog to digital signals Converting aplitude of input/output signals.

22 ,2. Hardware Pneuatic Valves In ineral processing operations where flow of fluids and slurries are frequently easured, a valve at the inlet and outlet of operation is coon. In effect the valve serves as a part of the operation as distinct fro instruents which indicate the status of a particular operating condition. In several circuits it fors the final control of the syste. Therefore its transfor has to be included in block diagra algebra. Fig is a scheatic diagra indicating the principles of operation of a typical pneuatic valve. The flow through the valve is governed by the design of its shape and space allowed between the valve and the valve seat. The extent of flow is generally given by the ste position. Valves are shaped so that the flow through could have a linear, square root or hyperbolic relation to opening. Matheatically these can be expressed as: Linear Square root Hyperbolic F(x) = x F(x) = V F(x) = where x is the opening distance. In Fig it can be seen that the pneuatic pressure actuates the diaphrag which in turn affects the ste which operates the position of the opening anipulated by a plug. To deterine the transfor of the valve operation, let us assue that the area of the diaphrag is A, and that it is displaced by a distance 5 when a force (H 8) is exerted via a spring, H being the Hook's constant. A factional force, ji, is active in the opposite direction due to the tight packing to stop leaks between the ste and the body of the valve. The frictional force u, acting, say upwards, for a short tie t, would be u. ddvdt. Pneuatic pressure P Spring Inlet I Outlet Fig Scheatic diagra of a valve for control fluid flow.

23 644 Let P be the signal representing the force that opens and closes the valve. displaceent 8 of the valve (diaphrag) is equal to the opposing pressure of the spring. The M d 2 d dd PA= r + fi + HS (18.47) Dividing by H and transposing,, i lz i c _ (]R Afi^ This is a second order differential equation and as we can put: M 1,,. f = (18.49) g ti H 0 The standard transfor of this second order syste is: P(s) '' A [M/Hg\ / H ^ H, + H (18.50) The present trend is to invariably use electronically operated actuators instead of pneuatically operated valves, hi either case the transfor will reain unaltered [15] Other Hardware Other hardware relates to electrical functions (capacitance, resistance, and inductance) echanical systes like springs, frictional systes, rotational systes and echaniss to suit particular setups. Transfer functions of soe coon selected hardware are given in Appendix C Controls of Selected Mineral Processing Circuits Mineral processing operations are dynaic systes. Disturbances arise priarily due to variations in process inputs and also due to the process achinery both of which cobine to affect the sooth operation of processes. To understand the strategy of process optiization and control for steady operation let us consider a siple unit operation like the control of the solid level in bins and hoppers or the fluid level in a tank. The principle underlying the control strategy is the sae in both systes while the hardware and operation ethods differ. Level control of bins, bunkers and solids storages, are ore coplicated as the solids in these vessels are alost never level. For sooth liquid level in tanks, this proble obviously does not arise, hi the following section the sipler situation of controlling the fluid level in tanks or siilar vessel is exained Controlling Liquid Level in Tanks Let us consider a tank fed by a liquid at the rate of Fi. The tank level has to be aintained by an output F o. Let us assue that a level L (Fig ) has to be aintained in the dynaic state using a sensor to easure the hydrostatic pressure difference between the top and the

24 645 Feed F, hsp 1 Sensor Gc(s) h (s) Fig Liquid level control [11]. botto level of the tank. Let us also assue that a disturbance is applied to the feed syste and the output F o is the anipulative variable which is irreversible. A block diagra describing the process is given in Fig The operation involves a change in level L in tie t. If A be the crosssectional area of the tank which is continuously fed and eptied, the process ay be expressed atheatically as: dl (18.51) where Fi = load and F o = anipulative variable, (output). The transfer function of this first order differential equation is: As As (18.52) To exaine the steady state condition the transfer function of each of the coponent is required. For deterining the transfer function of sensor its operation has to be exained. We have already considered the sensor as a differential pressure transducer, where the differential pressure is generated by the pressures at the top and botto levels of the fluid in the tank. As the height of the liquid level changes it is obvious that AP also changes by an aount proportional to L, that is: AP = KL where K is a constant. If the change is brought about by the displaceent of the sensing diaphrag by an actuating pressure AP, then a force balance around the sensing diaphrag can be established. Let L be the displaceent of the diaphrag of the sensing device due to a change in level affected by a pressure differential AP and ti,, and kp are constants that are functions of the instruents. Then according to Stephanopoulas [11] the sensor operation ay be described by a second order differential equation as: dl dl dt = K p AP (18.53) And the transfer function of this second order equation is:

25 646 Eq. (18.54) therefore is the transfer function of the sensor. Let us assue that a P+I controller is used and a set point Lsp is set, Then the transfer function of the error, e will be: e =[lsp(s)l (s)] (18.55) We have seen (Table 18.2) that for a P+I controller whose output is O, the transfer function is given by: O(s) = G c 1 + e(s) (18.56) L Ts J For the final control eleent, that is a control valve, the transfer function of the response is a first order syste and therefore can be written as: F 0 (s) = ^c(s) (18.57) T+l Fro the block diagra it can be seen that the coponents operate in series. Thus Eq. (18 39) applies. In the general case, if Gc(s) is the transfer function of the controller, GF(S), that of final eleent, Gp(s), of the process, G that of the sensor (easuring device), then according to block diagra algebra the response of the output to the change of set point will be: G(s)G(s)G(s) l + G(s)G(s)G(s)G(s) G p F c SP And the effect on output due to a disturbance (change in load) will be: O, (s) =.. G, A y., x d(s) (18.59) 2 W l + G p (s)g F (s)g c (s)g (s) The closed loop response will be the su of Eqs. (18.58) and (18.59), that is: G P (s)g F (s)g c (s) G H O (s) =,, F Y, C Y x + T^ 7^ ^ TT xd 0) ) l + G P (s)g F (s)g c (s)) l + G p (s)g F (s)g c (s)g (s) The response in tie doain to a change in the level due to a change in load will be given by the inverse of Eq. (18.60) which would contribute to the control the level of tank.

26 647 The principles explained in this siple control syste are applicable to all controlling operations. In integrated unit processes encountered in noral ineral processing operations, the applications becoe ore coplicated and would depending on the circuit diagra. In the following, principle considerations which go to control selected ineral processing operations are described Crushing Plant Control The control of any crushing operations starts with the control of the feed syste. Lynch [16] has suarized the disturbances in a crusher as: 1. ore properties (size and hardness), 2. ore feed rate, 3. crusher settings (close and open settings) alterations due to wear and tear, 4. surge in feed load and plant power draw. The control loops of a gold crushing plant are shown in Fig [14]. Two regulatory control loops and a supervisory control loop are shown. The function of one regulatory loop (shown within dotted lines) is to control the feed tonnage on the feeder conveyor and the other to regulate the crusher power which in turn regulates the feed. The supervisory loop is a cascade loop which deals with any ibalance between the two regulatory loops through level changes in the feed hopper. The basic instruentation involved would include a transducer to onitor the closed side setting (and or open side setting), an A/D connection to the controller and a D/A connection to the level indicator [16] Grinding Mill Control in Closed Circuit The principle objective for controlling grinding ill operation is to produce a product having an acceptable and constant size distribution at optiu cost. To achieve this objective an attept is ade to stabilize the operation by principally controlling the process variables. The ain disturbances in a grinding circuit are; P+l Controller Proportional controller (P) P+l controller Manually set, Power set point Feed Level gauge fs.l/i Crusher Wo Fig Control of gold crushing plant [14].

27 change in ore characteristics (ore feed rate, grindability, feed particle size distribution, ineral coposition and ineral characteristics like abrasiveness, hardness), 2. changes in ill operating paraeters like variation of input flow rate of aterial like surging of feed caused by pups and level of ill discharge sup. The ill control strategy has to copensate for these variations and iniize any disturbances to the hydrocyclone that is usually in closed circuit. The siplest arrangeent is to setup several control loops starting fro the control of water/solid ratio in the feed slurry, sup level control, density control of pulp streas at various stages and control of circulating load. Presently ost ills use centrifugal pups for discharging fro the sup. This helps to counter surges and other probles related to puping. For feed control the ost likely option is to use a feed forward control while for controlling the hopper level and ill speed and other loops the PI or PID controller is used. The control action should be fast enough to prevent the sup fro overflowing or drying out. This can be attained by a cascade control syste. The set point of the controller is deterined fro the level control loop. This type of control prootes stability. As an exaple, for copletely controlling a grinding ill circuit the operation of a SAG ill is considered here as these ills see to be slowly displacing the noral ball ill operations. The SAG ill characteristics have already been entioned earlier in Chapter 9. The ain variables are: 1. solid ass transported through the ill ( solid feed plus the circulation load), 2. the ill discharge solids, and 3. the overflow solid flow. Fro the control point of view, the additional interests are: 1. overfilling of the ill, 2. grate restrictions and, 3. power draft. Each of these is controlled by specific controlled inputs, i.e., feed rate, feed water and discharge water flows. The overflow solids fraction is controlled by onitoring the ratio of total water addition (WTOT) to the solid feed rate. The ratio being fixed by the target set point of the overflow solid fraction. Usually the charge volue of SAG ills occupy between 3040% of its internal volue at which the grinding rate is axiized. When the charge volue is ore, then the throughput suffers. The fill level is onitored by ill weight easureent as ost odern ills are invariably ounted on loadcells. During the operation of SAG ills, it is soeties observed that the sup levels fall sharply and so does the power draft. This phenoenon is attributed to flow restrictions against the grate. When this occurs it is necessary to control, (or in extree circustances), stop the incoing feed. The power draft is the result of the torque produced by the ill charge density, lift angle of the charge within the ill and fill level. The relationship between these paraeters is coplex and difficult. Therefore to control ill operation by power draft alone is difficult.

28 649 For the purpose of stabilization of the circuit, the basis is to counteract the disturbances. Also the set points ust be held. The set points are attributed by dynaic ass balances at each stage of the circuit. In odern practice the structure and instruentation of the control systes of tubular grinding ills are designed to operate in three levels or in soe cases four levels. The control loops and sensors for a SAGill and the levels of control are illustrated in Fig According to Elber [10,17], the levels are: Level 1 The operation at Level 1 ainly consists of controlling the feed rate and the water inputs. In addition to this the SAG ill revolving speed and secondary circulating load also fors ancillary loops. There are four ain control loops in Level 1 (Table 18.4). Table 18.4 Control loop of SAG ill Control Loop Feed Flow rate Sup water flow Sup level (Discharge feed regulator) Control Variables Feeder Speed ( Feed otor speed) Valve position Level indicator The ain sensors are: 1. load cell for ill weight, 2. power easureent (aeters, volteters), and 3. density gauges (yray density gauge) for on line, noninvasive, easureent of slurry densities. Level 2 The function of Level 2 is to stabilize the circuit and to provide the basis of optiizing function in Level 3. Three cascade loops operating in level 2 controls that function in conjunction with level 1 controllers. The cascade loops are: 1. ill load feed rate (controlled by feeder speed), 2. ill discharge, (% solids), and 3. ill product, (% solids). The set points are supplied by level 3 controllers for all the cascade loops. The ill load and percent solids in the two streas are calculated fro signals received by sensors in the water flow strea, the sup discharge flow rate and the density readings fro density eters in the pulp streas. The ill load cells supply the charge ass. The load cell signals are copensated for pinion up thrusts [10]. The set points for the ill load and the two pulp densities are given by level 3 controls. The points ay also be set by neural ethod of analysis or fuzzy logic expert systes.

29 650 C11 Level 1 Level 2 Wt C12 C21 t C22 Load Copo. h Level 3 Load set Point Mill discharge % solids Set point Process Water C13 X C23 Product, % solids Set Point C14 C15 % solids Cop. Ws Wf Prod. %S Cop. Fig Three levels of control of a AG/SAG ill. To deterine the set point for the optiu ill load, a relation between load, consisting of different feed blends and perforance (the axiu achievable throughput) is established. Siilar observations are ade for ill discharge density and ill discharge flow. For soe routine standard situations ready ade supervisory work station coputer progras, especially for Levels 1 and 2, are available.

30 651 Level 3 The priary function at Level 3 is optiisation of the SAG ill operation. That is, control of the product at optiu level. In an integrated situation where ball ill and cyclone is in the circuit, the optiisation ust take place keeping in ind the restraints iposed by down strea requireents. This optiisation can best be achieved by developing a software for coputer use. Usually a large database is required to cover infrequent control actions Thickener Control The control of thickener operation is directed to obtaining a clear overflow as rapidly as possible. The sedientation rate is usually accelerated by additions of flocculants. Flocculants are added in the feed pipe (or feed launder) and axiu dispersion attepted by appropriate design of its entry to the thickener tank. The object is to entirely cover all the surface of ineral particles. The choice of flocculent and its concentration vary. It depends on the inerals present in the slurry, their coposition and their surface characteristics. It is necessary not to have too uch turbulence at the entry point of the tank.. For this purpose, the overflow level is kept sufficiently high above the feed level to ensure acceptable solid concentration in the overflow. Fig is a scheatic diagra of a thickener showing the different paraeters. Fro a process control point of view the design paraeters and ajor variables are: 1. height of the overflow clear fluid, H, 2. height of the bed level fro discharge end, HB, 3. solids inventory, S, 4. sold ass inflow, Si, 5. solids ass outflow, So, 6. average bed solid voluetric fraction, vs It can be seen that the solid inventory S = A HB VS ps, where ps is the solids density. Overflow hw Feed+Flocculent ) Bed level Fig Scheatic diagra of thickener.

31 652 At the bed level the solids residence tie, tr will be: tr = I (18.61) The dynaic solids ass balance is: ds (SS) (18.62) During operation, if the feed flow changes, i.e., increases or decreases, the flocculent input changes proportionately. A control loop in the flocculent charging device involving a pup is required to follow these changes at an appropriate level of control. The power, P, required by the pup, which is assued to be connected to a horizontal pipe with no bends, is given by: P = kn (F D ) 2 kw/h (18.63) where k = constant (proportional to pipe length), [i = viscosity, FD = underflow discharge flow rate. The pup pressure obviously varies as the solids ass outflow (kg/ 3 ) and according to Elber [10], is given by: P = r ^S Pa (18.64) 2 P B s Control Strategy The control of the solid contents in the overflow and underflow streas is the basis of thickener control. The average bed density (solids inventory) has to be controlled by the underflow flow rate and the flocculent additions to the slurry. To attain target overflow solids concentration the underflow density should be sufficiently high. This is obtained by longer residence tie of treated slurry in thickener. The underflow flow rate is easured by agnetic or ultrasonic flow eters. The control schee can now be suarized: Level 1: Control loops Two ain loops are placed in Level 1. The first ain loop (# 1) is for underflow control. The second ain loop (#2) is for the control of flocculent flow. At level #1, the essential process easureents for underflow control are: 1. rake torque with a torque eter fixed to the rakes, 2. bed level by using a siple float or vertical position sensor, 3. thickener bed pressure, by easuring the pulp pressure on the floor by a sensor.

32 653 At level #2 for flocculent loop control, easureents are chiefly flow rates of fluids by standard flow eters and power draft easureents for variable speed positive displaceent pup. Other easureents at this level include: 1. pup speed, 2. underflow density easureent (yray density gauge), 3. pup discharge pressure by standard pressure gauge. Level 2: Control loops The ai of Level 2 is to keep the underflow bed level close to target. The set point of the bed level is therefore controlled by the bed level controller. In practice it is found that the bed level can be disturbed by high bed density which could result in high torque on the rakes. To avoid such situations the basic flow control is designed to be overridden. This is achieved by providing a highselector that outputs the flow set point [10]. Such an arrangeent is shown in Fig where it can be seen that the bed density and rake torque with axiu liiting values are connected to the high selector. The output fro high selector or the flow controller set point has a set low liit. A safe flow is therefore aintained fro the underflow. The pup that pups the underflow is set to high and low speed liits taking signals fro the output of the flow controller The advantage of this syste is that in the event the thickener operation ceases due to say, stoppage of ill operation, and therefore feed to the thickener, the underflow pup continues to operate till the thickener is epty and chances of clogging is reote. The pup speed controller incorporates liiting power draft so that the pup does not trip at high power For controlling the flocculent flow signals are taken fro the bed density controller. Controlling the flocculent flow is difficult by this ethod as it takes tie for the flocculent to properly ix with the rest of the inventory. The speed of response varies with the rate of change of bed density. Elber [10] suggests using underflow flow to "control inventory in conjunction with bed levelflocculent dosage cascade." Max Process Variable Max. Liit Process Variable ^ Bed Density controller Bed level controller High Selector Underflow Controller Pup Speed r Set Point Process Variable Rake torque controller Underflow Fig Underflow control setup [10].

33 654 Level 3: Control loops Level 3 control involves optiisation of thickener operation (and pipe lines). This includes cost function based on: 1. flocculent consuption, 2. pup power, 3. discharges to tailings. All these factors depend on the underflow density set point. Optiu conditions are usually ascertained by trial and error ethod by taking signals fro the underflow density, pup discharge pressures and pup power drafts and estiating the corresponding cost functions Control of Hydrocyclone Operation During steady operation the products fro a hydrocyclone has a definite cut point. However due to variations in the feed slurry characteristics and changes in the hydrocyclone geoetry, especially the diaeter of the apex due to abrasion, the cut point changes during operation. It is necessary to hold the perforance at the desired dsoc value for down strea operations. The control strategy could be to onitor the deviation of the cut point. The alteration in cut point was obviously due to change in feed characteristics and additionally to changes in cyclone geoetry due to abrasion. Working with a D6B Krebs hydrocyclone and quartz suspension of known particle size distribution Gupta and Eren [18] indicated that for a constant pressure differential, the relative effect on dsoc was: Du>(p, >Q>H>T where Du = Diaeter of apex, c., (pi = Voluetric fraction of solids in the feed slurry, Q = Rate of flow, 3 /in, H = Height of the cyclone, c, T = Teperature, C. The logic of the control progra adopted was to calculate the dsoc value during a steady state condition using a atheatical odel. When the cutpoint was altered due to any change in the variables the coputer sequentially searched for the offending variable and restored it to the original value. The restoration was done by iteration using perturbance technique. The advantage of the technique was to predict changes using the previous reading as the initial value. Thus a variable (Du, <p, Q or T) was chosen by the coputer and the established odel was considered as /(x) and the step changes in dsoc calculated using the expression: Ad 5 oc dx (18.65) where x expressed the variables. After repeated iteration when the set value of dsoc was achieved, or tended to zero, the iterations ceased. A PID controller was adequate.

34 655 An iportant factor in designing control loops is the instruental and prograable tie delays, hi the case of hydrocyclone autoation, the sources of tie delays is given in Table Each instruent has to have a separate tie delay factor which could be up to 3 seconds Prograable tie delays introduced during iteration could be greater than instruental tie delays Table 18.5 Source of tie delays in a hydrocyclone circuit. Equipent Motor pup set Vortex finder positioner Spigot diaeter Density gauges and Flow eters Coputer Tie delay source Frequency controller, Inertia in otor load. Vfl and I/P conversions, Pressure transission, Mechanical oveents, Servo Mechanis operations V/I and I/P conversions. Electronic, Adjustable response tie Conversions Fig shows the set up and instruentation for autoatic control of a Krebs D6Bhydrocyclones. The apex of the cyclone was fitted with a rubber sleeve which could be pneuatically squeezed to alter its diaeter. The vortex finder was specially designed to travel up and down. The centrifugal pup was fitted with a fiequency controller. The control strategy is illustrated in Fig Laboratory trials suggested that when a hydrocyclone variable was subjected to a step change and the dsoc value deviated, the operation of the hydrocyclone could be restored such that the dsoc could be aintained to within ± 5 % of the calculated value. The atheatical odel used for calculations was derived to suit specific slurry conditions. The conclusion was that such techniques could be developed for autoatic control of the cut point of hydrocyclones , Advances in Process Control Systes Mineral processing operations involve a nuber of process variables that change randoly with uncertain frequencies. The control strategies developed with the use of PID controllers have been found to be inadequate especially in nonlinear systes and systes with large lag ties. The present developent to solve these probles fall under two categories: 1. developent of algoriths for dynaic odels involving coplex statistical approach, 2. adoption of "Expert Systes" where atheatical odelling appear to be inadequate.

35 656 Vortex finder positioner 20100KPa Hydro cyclone 600 KPa Pup Pneuatic Valve Positioner Motorcurrent Inlet F/ yray D/g Outlet F/ Diff P/g yray D/g Pneuatic servopositioner Frequency Controller 420 A, 210 V 420A, 010V A/D A/D A/D A/D A/D D/A D/A D/A COMPUTER F/ = Magnetic Flow eter, D/g= yray Density gauge, P/G= Pressure gauge, D/A = Digital to analog transducer, A/D =Analog to digital transducer Fig Basic instruentation for autoatic control of hydrocyclone [18]. The algoriths developed and used in practice for nonlinear systes are: 1. self tuning control (STC), 2. extended horizontal control. In soe cases new filtering strategies have been introduced, like Kalan filter [1921]. These filters take care of variables that cannot be easured easily (like noise) Self Tuning Control (STC) The self tuning control algorith has been developed and applied on crusher circuits and flotation circuits [2224] where PID controllers see to be less effective due to ieasurable

36 657 Select Du, Q, and H SetQ, Du and H Input Slurry Teperature Wait for steady flow rate (sees) Scan instruents and calculate averages of Q, Qo, qj o, <p, and AP Use the odel to calculate d~ Calculate d^c *, d 5 oc,,,i,,, using Du 1118X, Du, nin, H rrsl,, H,,,,,,, Q rrlils, Q, ll Print djocraas and dsocin Select d5ij[. value d3ii^rt1it1< d.^n;r< dsin.:ax Yes > Calculate Ad 5 rx: = djoci dsoc Yes Adjuf within ± 5% of d VK: No Reset H by using H + K,*Ad 5, Calculate d^o^ theoretical using perturbation technique Yes Yes nas or Il1Miii attained No Wait for steady flow (sees) No Scat] instruents and calculate average readings ^ 1 " No i cs Exit Exit n=.16o Yes Calculate d?oc using instr. reading Yes Calculate d^uc (theoretical) using perturbation Du iii attained End Yes Reset Q using Q + K 3 *Adan: Calculate Ad50c {theoretical) using perturbation Qniax or Qiii attained No Scan instruents and calculate average readings I 1 No n=360 ^ (Pi within ± 1% of prev. tp L y~ Yes Calculate dw< using instr. readings Wait for steady flow rate (sees) Scan Instruents and calculate average readings <p ; within ± 1% of prev. (p L Yes Calculate d 5 oc using instr. readings n=36o T, Fig Strategy for autoatic control of d 5O c in a hydrocyclone.