A Regulator for Continuous Sedimentation in Ideal Clarifier-Thickener Units

Transcription

1 A Reglator for Continos Sedimentation in Ideal Clarifier-Thickener Units STEFAN DIEHL Centre for Mathematical Sciences, Lnd University, P.O. Box, SE- Lnd, Sweden Abstract. The prpose of this paper is to present a reglator for control of the continos-sedimentation process in a clarifier-thickener nit when this is modelled in one space dimension and when the settling properties of the solids obey Kynch s assmption. The model is a scalar hyperbolic conservation law with space-discontinos flx fnction and point sorce. The most desired type of soltion contains a large discontinity. A common objective is to control the movement of this discontinity sbject to the reqirement that the efflent of the process have zero concentration of particles. In addition, there may be a reqirement that the nderflow concentration of the thickened sspension lie above a predefined vale. Based on previos reslts on the nonlinear behavior of the process, a nonlinear reglator is presented. It controls the location of the large discontinity indirectly by controlling the total mass. The process is stabilized significantly and large inpt oscillations can be handled. Keywords: continos sedimentation, control, nonlinear reglator, dynamic behavior, clarifier-thickener. Introdction The aim of the process of continos sedimentation is to separate particles from a liqid in a large tank nder a continos inflow of mixtre at an intermediate feed level. The particles settle by gravity and are also inflenced by a blk flow pwards above the feed inlet the clarification zone), and a blk flow downwards below the feed inlet the thickening zone), see Figre left). Under optimal operating conditions, there is a discharge of a highly concentrated sspension at the bottom the nderflow) simltaneosly with a clarified overflow of liqid at the top of the tank the efflent). The continos-sedimentation tank is widely sed in mineral processing, wastewater treatment plants, chemical engineering etc., and is called clarifier-thickener nit, or settler. Under optimal operating conditions there are no particles in the clarification zone and a large discontinity in the thickening zone, called the sldge blanket in wastewater treatment. This state of the settler is called optimal operation. The process has been sed for abot a centry and is well known to be nonlinear, which is why its behavior is difficlt to predict as well as to model. The need to control the process for obtaining a clarified efflent is obvios. In a wastewater treatment plant most of the concentrated nderflow, which is biological sldge, is recycled within the plant to a biological reactor that precedes the settler. Therefore, it is also vital to be able to control the nderflow concentration. In sch an activated sldge system the settler also serves as a bffer of biological sldge. These aspects can be flfilled by controlling the sldge blanket level. Independently of the application, the process is highly nonlinear even nder the most common idealized assmptions, which are the following. The clarifier-thickener nit is ideal in the sense that all flows in the tank occr only in one dimension, the feed inlet is a point sorce, the cross-sectional area is constant and the concentration is constant on each cross-section. Frthermore, the particles are assmed to be eqally-sized spheres that form a non-compressible sediment at a maximm concentration. c 7 Klwer Academic Pblishers. Printed in the Netherlands. SDJEM.tex; //7; :; p.

2 S. Diehl In the series of papers [ ] thorogh investigations have provided a deeper knowledge as well as classifications of the nonlinear behavior of an idealized one-dimensional clarifierthickener model. The hyperbolic PDE-model was formlated and analyzed in [, 6], in which existence and niqeness locally in time were proved. Global existence and niqeness were established by Bürger et al. [7, ] and Karlsen and Towers [9]. The PDE-model is hyperbolic becase of the constittive assmption by Kynch []: the settling flx of particles is a fnction only of the concentration. We refer to the series [ ] for jstifications, discssions and references regarding the present model as well as the parallel engineering development withot PDEs. We mention only the recent important contribtions by Bürger et al. [, ], which rely on the analyses by Karlsen et al. [, ]. They formlate and analyze a more general PDE model which incldes compression at high concentrations. Analyses, based on PDE soltions, of the possibilities of controlling the sldge blanket large discontinity) can be fond in [ ]. In all these references it is assmed that the variation of the feed inpt is so moderate that a qasi steady-state sitation remains with a sldge blanket. The limitations for control are analyzed by the athor in []. An interesting simplified, lmped parameter model was presented in a short bt comprehensive paper by Stehfest [] in 9. In contrast to several other pblished models not referred to here), the argments behind Stehfest s model, in particlar considering the bondary conditions, agree with the theory of conservation laws with space-discontinos flx fnctions that was developed dring the 99 s [,, 6, ]). The step responses presented in [] agree with those in Section. in the present paper. Chancelier et al. [9, ] se a feedback law to control the sldge blanket. However, it reqires that the sldge blanket level is measred, which may be difficlt. In the present paper we show how the sldge blanket can be controlled withot measring its location. The need to control the settler is also emphasized, directly or indirectly, in the applications, see e.g. [ 6]. For step inpts, optimal control strategies were presented in [] in order to meet the different control objectives sggested. Those control objectives are exhastive in the sense that they can always be met, also for theoretically possible bt maybe nrealistically high vales of the feed variables. In [], it was shown there how the process cold be controlled, however, not atomatically. There is a need for a refined control strategy for fine-tning the sldge blanket level. This is taken care of in the present paper. In Section, the process, the model and the previos reslts are reviewed briefly. The main condition of the previosly presented control objectives to maintain optimal operation as long as possible is now refined to inclde control of the sldge blanket, possibly with a constraint on the nderflow concentration, see Section.. Section contains jstifications and reasons behind the control strategy, which can be fond in Section.. The strategy means that the sldge blanket level is controlled indirectly by controlling the mass in the settler. The strategy is realized by a proportional reglator in Section, which works for moderately varying indata. For indata with large oscillations, the proportional reglator has to satrate according to the limitations of the control variable presented in []. The complete nonlinear reglator can be fond in Section, which concldes with several simlations illstrating some properties of the reglator. SDJEM.tex; //7; :; p.

3 A Reglator for Continos Sedimentation. Preliminaries We review only briefly the fndamental notation and reslts presented in the papers [ ]. These concepts are sfficient for nderstanding the ideas, reslts and simlations of the paper. For the details of constrction of the soltions shown in Section., we refer to [, Section ]. For the nmerical simlations we se the data and batch-settling flx fnction given in the caption of Figre and the nmerical method in [7]... THE CLARIFIER-THICKENER UNIT AND THE MODEL The one-dimensional model of the clarifier-thickener nit, or settler, was first presented in [6]. Figre shows the settler and the flx fnction in the thickening zone for three different vales of the control parameter Q. The prposes of the settler may vary depending on in Q f f Q e e clarification zone H f b ) + q thickening zone f) = f b ) + q f b ) + q Q D x 6 M infl m M max Figre. Left: Schematic pictre of an ideal one-dimensional clarifier-thickener nit, where stands for concentration and Q for volme flow of the feed, efflent and nderflow streams, respectively. The flow restrictions are Q f = Q e + Q > and Q e. For the nmerical simlations we se H = m, D = m, and A = π m 7 m for the constant cross-sectional area. Right: Flx crves f) in the thickening zone and characteristic concentrations. The blk velocities are defined as q e = Q e/a etc. The constant infl is the inflection point of f b ) and f) = f b ) + q. For q < q < q there is a local minimm point M of f) that lies between infl and max. Given M, m is the lower concentration defined by f m) = f M). For q < q there is a local maximm point, M < infl ) of f). The batch-settling flx sed for demonstrations with nmerical simlations is f b ) =.6/ max) ) [ kg/m h) ]. what indstrial process it is involved. At least in wastewater treatment the main prposes of the settler are the following. It shold. prodce a low efflent concentration;. prodce a high nderflow concentration;. work as a bffer of mass and be insensitive to small variations in the feed variables. The one-dimensional model of the settler is the following. The conservation law can be written as the partial differential eqation t + F,x,t) ) = st)δx), ) x SDJEM.tex; //7; :; p.

4 S. Diehl where δ is the Dirac measre, the total flx fnction is q e t), x < H g,q e t) ) = f b ) q e t), H < x < F,x,t) = f,q t) ) = f b ) + q t), < x < D q t), x > D, and the sorce fnction is st) = Q ft) A ft) = Q t) + Q e t) f t) = q t) + q e t) ) f t). A For convenience, the dependences of the flx fnctions within the settler on the time varying) volme flows are only written ot when it is needed, i.e., f) = f,q t) ). The physical inpt variables are the feed concentration f and the feed volme flow Q f. For graphical interpretations in operating charts it is, however, convenient to se the feed point f,s) as inpt variable. The control variable of the process is Q and has the natral restriction < Q Q f. Two particlar vales of this variable arise from the properties of the batch settling flx fnction. Define q = f b max ), Q = q A, q = f b infl ), Q = q A, which are the blk velocities sch that the slope of f is zero at max and infl, respectively, see Figre right)... OPERATING CHARTS AND OPTIMAL OPERATION Figre shows the steady-state chart and the control chart. Depending on the location of the y f M ) l O O O l l l p U y Λ P P Λ b Λ Λ a U Λ l m M f M) q max 6 Figre. Left: The steady-state chart. The thick graph is the limiting flx crve. If the feed point lies on this crve, the settler is critically loaded in steady state, which means that it works at its maximm capacity. Below this graph the settler is nderloaded, and above it is overloaded with a non-zero efflent concentration. Each region corresponds to a specific steady state which is niqe, except on the limiting flx crve and on l and l ), where the location of a discontinity in the thickening and/or the clarification zone is not niqely determined. Note that the regions in this chart all depend on Q. Right: The control chart with respect to steady states; Λ = Λ a Λ b, Λ = Λ Λ. The regions in this chart are fixed given the batch settling flx f b ). SDJEM.tex; //7; :; p.

5 A Reglator for Continos Sedimentation feed point f,s) in the steady-state chart, there are different possible steady-state soltions, which are all piecewise constant and non-decreasing with depth; see [, Table ] for a complete table. The limiting flx is defined as: f lim ) = min fα) = α max { f), [,m ] [ M, max ], f M ), m, M ), see Figre left). This flx, as well as the characteristic concentrations and the regions of the steady-state chart, depend on the control variable Q ; e.g. M Q ), f,q ) and f lim f,q ). The following regions in the operating chart are independent of Q : Λ i = Q > l i Q ), i =,...,, P = P P, where P = pq ), P = pq ), <Q Q Q > Q Λ a = Λ { },y) : y < f b infl ) + q infl and Λ b = Λ \ Λ a, see Figre right). Given a feed point in this chart, there is a niqe graph f lim, Q ) that passes throgh the feed point, see [, Theorem ]. With this niqe vale Q on the control parameter, the settler is critically loaded in steady state, which means that any higher load mass per time nit) fed to the settler will reslt in an overflow of particles. A more important concept than critically loaded is optimal operation. These concepts are related bt not identical. Optimal operation in steady state means that the concentration is zero in the clarification zone and there is a discontinity in the thickening zone between the concentrations m and M, see Figre right). This discontinity is, in wastewater treatment, called the sldge blanket and its location at the depth x = x sb,d) is called the sldge blanket level SBL). A rising SBL refers to reality, althogh the x-coordinate decreases, becase of the downward-pointing x-axis. A necessary condition for this state is that f,s) pq ) l Q ) l Q ) and Q < Q, which implies f,s) P Λ Λ a, see Figre. For a general dynamic soltion, optimal operation and the SBL are defined as follows. Let cl denote the restriction of the soltion to the clarification zone. DEFINITION.. The settler is said to be in optimal operation at time t if Q t) < Q and the soltion of ) satisfies: cl x,t) = x,t) =, H < x <, there exists a level x sb t),d) sch that { [,infl ), < x < x sb t) x,t) [ infl, max ], x sb t) < x < D. The definition implies a natral definition of the SBL for a settler in optimal operation: it is the discontinity at the depth x = x sb t) in the thickening zone, sch that the jmp in the concentration passes the characteristic concentration infl. It is convenient to se this definition of the SBL also when there are particles in the clarification zone. SDJEM.tex; //7; :; p.

6 6 S. Diehl In the analyses of step responses and control of these, it trned ot that similar lines to the graph of the limiting flx were convenient to introdce. With the same notation as in the previos papers, we define and skip L since we do not need it here): { L = l i p,y) : y = q, f } M) < max, q i= L = {,y) : y = f ) } f), M where f ) = f M ), M < fm ) q f q, M ) q < max. Note that these sets depend on Q, see Figre. By the control strategy DCL direct control L D D S 6 l l S l L 6 S y = q y = q 6 M infl m M max 6 M infl m M max Figre. The set S = S S the safe region) is the closed region below L, shaded in the figre. D the dangeros region) is its complement, i.e. it lies strictly above L. Note that the feed point has to lie on or above the line y = q, since s = Q f f /A Q f /A = q f.) with respect to L ) we mean that Q t) is defined sch that f t),st) ) L Q t) ) analogosly for L ). Since the vale of Q is niqely determined by the feed point f,s), it is convenient to se the notation Q = L f,s) f,s) L Q ). To satisfy the three prposes of the settler mentioned above, some control objectives for the process were introdced in [, Table ]. The main condition of these is to maintain optimal operation as long as possible. From the analyses of step responses in [] and dynamic soltions in [], it trned ot to be convenient to introdce the following sets of the operating chart: S = {,y) : q < y f ) }, D = {,y) : max,y > f ) }, { } S = S,y) : y f infl ), S = S \ S, see Figre. For step responses from optimal operation in steady state, the state of optimal operation is left immediately if and only if f,s) D. For a general soltion, the sitation is slightly different. A sfficient condition for maintaining optimal operation, at least for a while, is f,s) S. SDJEM.tex; //7; :; p.6

7 A Reglator for Continos Sedimentation 7.. THE CONTROL VARIABLE S INFLUENCE ON THE UNDERFLOW CONCENTRATION Since one of the prposes of the settler is that the nderflow concentration shold be high, a natral constraint as a part of a control objective is that t) is bonded below. This can be described in terms of the control variable a priori, see the following theorem the proof can be fond in []) and Figre. THEOREM.. Assme that the settler is in optimal operation for t T. The nderflow concentration satisfies t) ū, max ] for t T, where ū = f infl, Q )/ q. Let min ū, max ) be a given desired lower bond on the nderflow concentration. Assme that Q t) Q max, where Q max is defined niqely by Then t) min f M Q max ),Q max ) Q max = A min. for t T and Q max > Q. Λ P Λ P Λ y = q y = Qmax A Λ 6 Q Q max min ) 6 infl ū min max Figre. Left: The characteristic concentrations of Theorem. can be obtained graphically in the operating chart for control of steady states in the following way. ū = 7. kg/m and Q = 9 m /h satisfies f infl, Q) = q ū. Given min ū, max) determine the corresponding y-vale on the bondary of Λ and Λ. This flx vale is eqal to Q max min /A. Right: The graph of Q max as a fnction of min. ū max. Control objectives and strategies.. CONTROL OBJECTIVES From the reslts in [] we know the limitations of the control variable Q t) for maintaining optimal operation in the sense that overflow is prevented and, if a control objective reqires it, the nderflow concentration is kept above a prescribed level. However, to maintain optimal operation dring a long time, it was also illstrated that there is a need to fine-tne the average SBL so that it stays within the thickening zone. The control objective optimal operation is SDJEM.tex; //7; :; p.7

8 S. Diehl maintained as long as possible, possibly sbject to a lower bond min on the nderflow concentration, needs therefore to be refined. Given a fixed reference vale x r sb of the SBL and a lower bond min, we introdce the following control objectives with respect to the SBL : COSBL: Optimal operation is maintained and x sb t) is close to x r sb. COSBL: Firstly, t) min holds, secondly, optimal operation is maintained and x sb t) is close to x r sb. The phrase close to cold mean, for example, that x sb t) lies in an interval arond x r sb. A frther natral reqirement is that x sb t) x r sb shold tend to zero if the feed point is constant after a certain time point. Another reqirement dring periodically varying inpt data cold be that the integrated absolte deviation dring a period shold be kept small. Then the control variable can be piecewise constant, which cold be another constraint. In order to formlate a control strategy for either of the two control objectives above, the control variable s inflence on the SBL as well as the nderflow concentration shold be known. The latter relation was presented in Section.. The former relation is considered in the next section... DISTURBANCES FROM OPTIMAL OPERATION For all transient soltions presented in [, ], it can be conclded that the following property holds: When the settler is in optimal operation, the concentrations above and below the SBL are sally approximately m and M, respectively. This means that a control strategy that scceeds in meeting COSBL or, yields a dynamic soltion that is approximately a stationary optimal-operation soltion. Therefore, to elaborate sch a control strategy, it is of vital importance to have information of the responses of the process to distrbances when the settler is in optimal operation in steady state. The relation between Q t) and x sb t) is difficlt to obtain generally. In fact, there is no niqe relation, since the actal concentration distribtion in the settler plays a role. From [] we know that given that the triple f,s,q ) implies a steady-state soltion in optimal operation, the soltion is niqe except for the location of the SBL, which can be anywhere in the thickening zone. Hence, the only relation in sch a case is that x sb is constant as long as Q is and the feed point is constant). However, given the location of x sb we can establish the response to a change in Q. Assme that the settler is in optimal operation in steady state. Then the feed point f,s ) lies on the horizontal straight part of L Q ), see Figre. We investigate for distrbances of this state sch that the feed point ends p above or below the horizontal line. We are only interested in small distrbances sch that optimal operation is not left directly. A feed-point step sch that f,s) lies below L Q ) After a step change in the feed point to f,s) U Q ), which means that s < s, the SBL is constant for a while and then declines. The mass decreases linearly). This was shown in [, Section : case U ]. A control-variable step p sch that f,s ) lies below L Q ) As the control parameter jmps p to Q > Q sch that f,s ) U Q ) holds, the soltion in the thickening zone is qalitatively as the one constrcted in Figre. A simlation is SDJEM.tex; //7; :; p.

9 A Reglator for Continos Sedimentation 9 shown in Figre 6. We can conclde that the SBL declines, i.e. x sb t) increases. The incoming s f f H m = x sb M 6 m f M D x t Figre. The case when f, s ) lies below L Q ) after a step p of the control variable. The flx fnctions left) are f ) = f, Q ) and f) = f, Q ). The soltion right) consists of three different concentrations in the thickening zone, separated by discontinities. Thin lines are characteristics. concentration x,t) 6 t axis x axis Contors of x,t) Figre 6. A simlation in the case when f, s ) =,.) lies below L Q ) after a step p from Q = to Q = 9. The nderflow concentration steps down from = 7.96 to t) = 6.9 for < t <. mass per time nit, Q f f = As = Q, is nchanged. Despite the fact that <, the otgoing mass per time nit, Q is greater than Q, since the mass decreases. The SDJEM.tex; //7; :; p.9

10 S. Diehl latter follows from the fact that for each x, D) the concentration is non-increasing with time. A feed-point step sch that f,s) lies above L Q ) From the step responses in [, Section : cases O a, O a, l a, U a ], in which s > s, it can be conclded that the SBL is constant for a while and then rises. The mass increases linearly). A control-variable step down sch that f,s ) lies above L Q ) As the control parameter jmps down to Q < Q sch that f,s ) belongs to the region between the lines L and L in Figre. The soltion in the thickening zone is qalitatively as the one constrcted in Figre 7. A simlation is shown in Figre. We can conclde that the s 6 f f H m x sb = M m f M M D x M t Figre 7. The case when f, s ) lies above L Q ) after a step down in the control variable. Flx fnctions left) and the soltion right). The bottom concentration is M for t >. The SBL rises, first with a constant speed, then with an increasing speed as the SBL is a contact discontinity), and then with a constant speed. SBL rises, i.e. x sb t) decreases. The incoming mass per time nit, Q f f = As = Q, is nchanged. Despite the fact that >, the otgoing mass per time nit, Q is less than Q, since the mass increases. This follows from the fact that for each x,d) the concentration is non-decreasing with time. A fndamental property All for cases above with small step-distrbances of a soltion in optimal operation show the following fndamental property, which is well known among all operators of clarifierthickeners e.g. []): Q t) decreases or st) increases = mt) increases and the SBL rises, Q t) increases or st) decreases = mt) decreases and the SBL declines. A control strategy mst take this fndamental property into accont. We can note from the for cases that the inflence on the mass is direct, whereas there may be a time delay before the SBL changes. Frthermore, the SBL may be difficlt to measre in a plant, particlarly dring transients. Since the concentrations and flows of the inpt and otpt streams can be measred, the total mass in the settler can be calclated. Or control strategy will be to control the total mass by SDJEM.tex; //7; :; p.

11 A Reglator for Continos Sedimentation concentration x,t) 6 t axis Contors of x,t) x axis.... Figre. A simlation in the case when f, s ) =,.) lies above L Q ) after a step down from Q = to Q = 97. The nderflow concentration steps p from = 7.96 to t) = 9.69 for t >. sing a reglator. To ensre that sch a reglator satisfies the control objectives, we need a relation between the mass and the SBL... THE STEADY-STATE RELATION BETWEEN THE MASS, THE SBL AND THE CONTROL VARIABLE For optimal operation in steady state the following relation holds between the mass, the SBL and the control variable: m ss x sb,q ) = A x sb m Q ) + D x sb ) M Q ) ) = = A D M Q ) x sb M Q ) m Q ) )), ) where Q = L f,s). A three-dimensional graph of this fnction is shown in Figre 9. Note that, for fixed Q, m depends affinely on x sb. From [] we know that given that the triple f,s,q ) implies a steady-state soltion in optimal operation, it is niqe except for the location of the SBL, which can be anywhere in the thickening zone. This implies that there is no relation between Q and x sb in ). Another interesting thing regarding the control problem is the following. For fixed x sb, the mass is a weighted average of the two concentrations m and M. Especially, as the SBL is in SDJEM.tex; //7; :; p.

12 max S. Diehl mssxsb, Q)/A m ss D/, Q )/A Q 6 6 x sb Figre 9. Left: The relation between the mass in the settler normalized by A), the SBL and the control variable when the settler is in optimal operation in steady state. This fnction depends on the batch settling flx f b. Recall that D = m. Right: The relation as x sb = D/ = m. Q Q the middle of the thickening zone, x sb = D/, the weights are eqal and m ss D/,Q ) = AD mq ) + M Q ) Since m Q ) infl and M Q ) infl as Q Q, m ss D/,Q ) is approximately constant for the given batch flx fnction, see Figre 9 right). Dring dynamic operation ) does not hold. However, m ss xsb t),q t) ) seems to be a fairly good approximation of the mass mt), becase of the above-mentioned property that a controlled settler in optimal operation is approximately in optimal operation in steady state... A CONTROL STRATEGY From [, Table ] we can conclde that a necessary condition for keeping optimal operation after a step inpt is that f,s) P Λ Λ a, see Figre. If, in addition, the SBL is not too close to the bottom ineqality 9) in [] holds), optimal operation can be maintained. Frthermore, if the SBL meets the bottom, it was shown that the SBL can be restored within the thickening zone again after a finite time. Accordingly, a necessary condition for maintaining optimal operation dring long time of dynamic operation is that f t),st) ) P Λ Λ a. ) Assming this holds we define the reference vale of the control parameter either as Q r t) = L f t),st) ) st) = f M Q r t) ),Q r t) ) ) or Q r t) = L av f t),s av t) ), where av f t) =. t+t t f τ)dτ, ) SDJEM.tex; //7; :; p.

13 A Reglator for Continos Sedimentation analogosly for s av ) for some positive nmber T, preferably the period in the case of a periodic inpt. Gided by ) we then define the reference mass: m r t) = m ss x r sb,q r t)). 6) Becase of the two relationships ) and 6) the absolte difference x sb t) x r sb is small if and only if m ss xsb t),q r t)) m r t) is small. Combining this with the abovedescribed property that mt) is approximated by m ss xsb t),q r t) ) in optimal operation, we conclde that a control strategy shold keep mt) m r t) small. The fndamental property of Section. yields the first part of the following control strategy: define Q t) sch that Q t) Q r t) = h mt) m r t) ) for some increasing fnction h with h) =. optimal operation is maintained, and, for COSBL, the nderflow concentration is bonded below. The first item is achieved by a proportional reglator, see the next section. The second item is achieved by adding satrating bonds, which are obtained from the reslts in [] and presented below in Section.. A proportional reglator.. THE REGULATOR The first item of the control strategy above is in this section implemented in terms of a proportional reglator. Assme that optimal operation holds for t >. Given the initial mass m in the settler at t =, the mass at time t is given by which is eqivalent to t mt) = m + Qf t) f t) Q t) t) ) dt, t >, 7) dm dt = Ast) Q t) t), t >, m) = m. Note that mt) is continos and piecewise differentiable, since s, Q and are piecewise C. For a constant K > we introdce the proportional reglator Q t) = Q r t) + K mt) m r t) ). 9) Sbstitting 9) into ) yields the linear, time-varying eqation dm dt + K t) mt) m r t) ) = Ast) Q r t) t), t >, m) = m. This can be integrated to ) SDJEM.tex; //7; :; p.

14 S. Diehl mt) = m e KUt) + where t + ) K τ)m r τ) + Asτ) Q r τ) τ) )e K U t) U τ) dτ, t >, ) U t) = t τ)dτ. Since optimal operation is assmed to hold, Theorem. implies ū < t) max, and we have the bonds < ū t < U t) max t for t >. ) Hence, the first term on the right-hand side of ) tends to zero exponentially as t. Assme now that the feed point remains constant after a certain time as well as the reference vale Q r, which may or may not be chosen according to ). Then the reference mass m r, defined by 6), is also constant. With the constant r As/Q r, ) can be redced to mt) = m r + m m r )e KUt) + Q r t r τ) ) ) U e K t) U τ) dτ. ) The second term on the right-hand side tends to zero exponentially by ). Assming that the soltion converges to a stationary soltion in optimal operation then the last term in ) tends to a constant, which is zero in the case Q r is defined by ), as t. This can be proved in the following way. For given f,s) the vale of the control parameter for a corresponding soltion in optimal operation is niqe, see [, Theorem ]: Q ) lim t Q t) = L f,s). Hence, 9) implies that m ) = m r + K Q ) Q r ). ) Frthermore, for the stationary soltion in the limit, ) yields m ) = A D M Q ) ) x sb ) M Q ) ) m Q ) ))), ) from which x sb ) can be calclated. Combined with ) we get x sb ) = D M Q ) ) A m ) M Q ) ) m Q ) ) = D M Q ) ) A m r + K Q ) Q r ) ) M Q ) ) m Q ) ). ) If, and only if, ) is sed to define the reference vale Q r = L f,s) = Q ), ) implies m ) = m r and ) becomes m r = A D M Q r ) x sb ) M Q r ) mq r ))). This affine relationship between m r and x sb ), together with the corresponding one 6) between m r and x r sb, yields x sb ) = x r sb. We sm p the reslts. SDJEM.tex; //7; :; p.

15 A Reglator for Continos Sedimentation THEOREM.. Given x r sb,d), arbitrary initial data and a constant feed point f,s) for t >. Assme that the reglator 9), with a given constant Q r, connected with ) implies that the soltion is in optimal operation and converges to a stationary soltion as t. Then the limit SBL is given by ), with Q ) = L f,s). In particlar, if Q r = L f,s) then x sb t) x r sb as t. Remark. The rate of convergence to zero of the second term of ) is exponential. In the case Q r = L f,s) it follows from the theorem and ) that t) r as t. The rate of convergence of this limit process is difficlt to establish since it depends on the soltion within the settler, which in trn depends on the control parameter vale from the reglator 9). Hence, the rate of convergence of the third term of ) is difficlt to obtain, and therefore likewise for the limit process x sb t) x r sb. Remark. Independently of K, Q t) L f,s) as t. Note that ) implies that m ) m r as K independently of whether Q r is chosen eqal to L f,s) = Q ) or not. If Q r Q ), then x sb ) x r sb holds generally as K, which can be inferred from ) and 6): x sb ) D M Q ) ) A mr M Q ) ) m Q ) ), K, x r sb = D MQ r ) A mr M Q r ) m Q r )... SOME PROPERTIES OF THE PROPORTIONAL REGULATOR We demonstrate the statements of Theorem. by considering a nmerical example. Initially, optimal operation holds with the SBL in the middle of the thickening zone; x sb = D/ = m. The feed point is f,s ) = kg/m,9.6 kg/m h) ) and Q = m /h. At t = there is a step change to f,s) =,.) OQ ) Λ. Withot any change in the control variable, there will be a rising SBL and an overloaded settler after a finite time, see Figre. When DCL is sed to define the new constant vale Q = 9 for t >, the response withot a reglator) is shown in [, Figres 7 ]. Optimal operation is maintained, bt the new SBL satisfies x sb ) > x sb. Connecting the reglator 9) with K = m /kg h), x r sb = m and Qr = L,.) = 9, the original SBL is restored, see the simlation reslt in Figre. A higher vale of K gives a more rapid convergence of Q t) and mt). However, there is a transient soltion in which the SBL may not converge mch faster. We demonstrate this by setting K = in Figre. Even if Q r is not defined according to ) the reglator may be of major importance to maintain optimal operation, cf. Theorem.. Assme that the reglator 9) is connected with the constant vale Q r = Q =. Then Q t) is continos, and hence also t). The simlation with K = in Figre shows a similar behavior as in Figre with the difference that the mass now converges to a slightly higher vale. According to Theorem., Q t) L,.) = 9 as t. Formlae ) and ) yield m ) =.7 tonnes and x sb ) =.7 m, respectively. Note that these two vales depend on the reglator gain SDJEM.tex; //7; :; p.

16 f ) 6 S. Diehl concentration x,t) 6 t axis Contors of x,t) x axis 6 Figre. A simlation of a step response reslting in an overloaded settler when no reglator is connected Q t) = Q = m /h). K. If K is chosen to a larger or smaller vale instead, the reglator implies different limit vales of the mass and the SBL, see Figres and Contors of x,t) Q t) 6 Underflow concentration 7. 6 Figre. A nmerical simlation dring hors of a step change to f, s) =,.) OQ ) Λ when the reglator 9) is connected with K =, x r sb = m and Q r = L,.) = 9 m /h. The constant reference mass m r t) =. tonnes, given by 6), is shown by the dashed line. The reglator restores the initial SBL, and the mass and control variable converge to its reference vales. SDJEM.tex; //7; :; p.6

17 y A Reglator for Continos Sedimentation Contors of x,t) Q t) 6 Underflow concentration 7. 6 Figre. A simlation sing the same data as in Figre bt with K = instead. Q t) and mt) converge faster, bt not x sb t) Contors of x,t) Q t) 6 Underflow concentration 7. 6 Figre. A simlation sing the same data as in Figre K = ), bt with Q r = Q = instead. The reglator implies that mt).7 tonnes and x sb t).7 m as t. Becase of the feed point jmp to,.) DQ ), and the fact that Q t) is continos, the settler is actally overloaded for small t > with some particles in the clarification zone. With the resoltion of the simlations, this can only be hinted in the contor graph in Figre, where the small K means a slggish Q t). The condition,.) D Q t) ) holds dring the time when Q t) < L,.) = 76, which is less than an hor in Figre. Compare the simlation in Figre with the second remark after Theorem.. The simlation shows that the high vale K = implies m ) m r in accordance with m ) m r as K. The simlation also shows that x sb t) converges to a vale close to x r sb, despite the theoretical condition x sb ) x r sb as K. Note that K corresponds to disconnecting the reglator and the settler will overflow. SDJEM.tex; //7; :; p.7

18 y S. Diehl Contors of x,t) Q t) 6 Underflow concentration Figre. A simlation sing the same data as in Figre, bt with K = instead. The reglator implies that mt).97 tonnes and x sb t).9 m as t. 6 Contors of x,t) Q t) 6 Underflow concentration 7. 6 Figre. A simlation sing the same data as in Figre, bt with K =. instead. The reglator implies that mt).9 tonnes and x sb t). m as t.. A nonlinear reglator.. THE REGULATOR To ensre that optimal operation is maintained, Q t) has to be partly less than Q by definition of optimal operation), partly not too small to avoid particles in the clarification zone. In other words, the proportional reglator 9) may satrate. Therefore, we introdce the following nonlinear reglator to satisfy the control objective COSBL or COSBL, see Figre 6. We assme that the feed point satisfies ) and that the mass is calclated continosly by 7). Firstly, define Q r t) by either ) or ). Secondly, for a given xr sb, set the reference mass m r t) according to 6). Thirdly, define Q t) = min Q max,max Q min t),q r t) + K mt) m r t) ))), 6) SDJEM.tex; //7; :; p.

19 A Reglator for Continos Sedimentation 9 f, Q f ) x r sb s = Q f f A 6) Q r ) or ) f, s) m r + proportional reglator 9) K + sat. bonds 7) 9) nonlinear reglator 6) Q continossedimentation process ) m 7) Figre 6. The closed-loop system of the clarification-thickening process with the reglator, feed forward and feedback loops. where Q max and Q min t) are satrating bonds, which are defined as follows and commented pon below. Firstly, set { Q max Q, COSBL, = Q max 7), COSBL. Secondly, let Q min t) satisfy either Q min Q min t) Q max and f t),st) ) S Q min t) ) theoretically safe) ) or Q min t) = min Q max,max Q min,l f t),st) ))) less restrictive) 9) where Q min is a small positive nmber. Considering COSBL, Q max shold not exceed Q by the definition of optimal operation. For COSBL we set Q max = Q max < Q in accordance with Theorem. to meet the constraint t) min. In a plant there may be other reasons for defining a lower pper bond, for example, a limited pmp capacity. In Definition. optimal operation), we have reqired Q t) < Q instead of Q t) Q, cf. 7). The difference is sbtle and of no practical importance. The only reason for the definition is that there exists no steadystate soltion with a discontinity in the thickening zone as Q Q, see []. Dring dynamic operation with a varying Q t), the soltion may satisfy all other reqirements of Definition. despite Q t) Q dring a bonded time period. SDJEM.tex; //7; :; p.9

20 S. Diehl The reasons for the two alternative definitions, ) and 9), of the lower bond Q min t) are given in []. The theoretically safe ) implies that optimal operation is not left. There are other less restrictive conditions for this, bt these reqire more information, for example, the actal concentration distribtion in the thickening zone. Hence, Q min shold be set to the smallest possible vale satisfying f,s) S Q min ). In many cases Q min can be chosen sch that f,s) lies on the horizontal bondary between S and S, which means that st) = f infl,q min t) ), see Figre. If the vale of s is so low that f,s) S Q = ), then we set Q min to a small positive vale Q min. Recall that we have assmed that Q t) >, since t) is ndefined as Q t) =. In a plant there may be other reasons for choosing Q min not too close to zero. A high vale on s implies a high vale on Q min, which may imply a fast declining SBL and a low nderflow concentration cf. Theorem.). Then the less restrictive condition that f,s) S may be advantageos. This yields namely the lower vale Q min t) = L f t),st) ) if this is positive; otherwise set Q min t) = Q min > ). The drawback is that there is an exceptional case, in which there are some particles in the lower part of the clarification zone dring a limited time period. This is believed to occr only rarely and the advantage of this lower vale of Q min is believed to be more important in the applications. Therefore, we prefer 9) in the examples below. Since either of the two minimm bonds described above may be greater than Q max defined in Theorem. to satisfy COSBL, the reqirement Q min t) Q max is inclded in both ) and 9). Hence, 6) implies that Q min t) Q t) Q max holds. Example. Assme that the initial data and step inpt are the same as in the example in Section., except for the location of the SBL, which is now close to the bottom. A simlation with the reglator 6) is shown in Figre 7. Dring the first 9 hors the mass is less than its reference vale and the control variable takes its lowest possible vale Q t) = Q min = 76 m /h. Any lower vale of Q t) wold imply particles in the clarification zone. The advantage of controlling the mass instead of controlling the SBL directly) is here illstrated clearly. After 9 hors the control variable converges qickly to its final vale Q r = L,.) = 9 m /h. Dring hors frther the SBL rises and then reaches its reference level... SOME PROPERTIES OF THE NONLINEAR REGULATOR By nmerical simlations we shall illstrate some properties of the reglator 6) given the periodic indata in [, Examples and ], see Figres and 9. In particlar, the inflence of the satrating bonds is demonstrated. Let x r sb = m and Qmin t) be defined by 9), with = m /h, in the following cases: Q min A. Q r t) is defined by ), K =, and Q max = Q to satisfy COSBL, B. Q r t) is defined by ) with T =, K =, and Qmax = Q to satisfy COSBL, C. Q r t) is defined by ) with T =, K =. and Qmax = Q to satisfy COSBL, D. Q r t) is defined by ), K = and Qmax = Q max to satisfy COSBL. Example, case A. In Figre, a simlation shows how the reglator inflences the periodic inpt in order to satisfy COSBL. Right after each jmp in the feed point, Q t) makes SDJEM.tex; //7; :; p.

21 A Reglator for Continos Sedimentation concentration x,t) 6 t axis x axis Contors of x,t) Q t) Underflow concentration 7. Figre 7. A -hor-simlation sing the same initial data and step inpt as in Figre, except for the initial SBL; x sb =.7 m. At t = there is a step change from f, s ) =,9.6) to f, s) =,.) OQ ) Λ. The reglator 6) is connected with K =, x r sb =, Q r = L,.) = 9, Qmax = Q = 9 and Q min = L,.) = 76. The two latter vales are shown by dashed lines. The reference mass m r t) =. is shown by the dashed line. COSBL is satisfied and the SBL is adjsted to the desired level after h. a large jmp and stays, dring a short time, at the satrating vale Q min t) = Q min = at t =,,,..., and Q max at t =,6,,... These large jmps are cased by the jmps in the reference vale Q r t) by ). Note that a jmp in Qr t) also implies a jmp in mr t); see 6). Example, case B. When defining Q r t) by ) with the time average taken over a period T = h), it will be constant and eqal to the initial vale Q. This corresponds to the initial feed point, which is the mean vale of the periodic inpt, see Figre pper left). Ths, Q r = Q = L.,7.) = and mr t) =.6. Then Q t) depends continosly on time nless it has to jmp becase of the satrating bonds. This is demonstrated in Figre, where small jmps in Q t) can be seen at t =,6,,... At each of these time points the SDJEM.tex; //7; :; p.

22 S. Diehl D S S 6 Feed concentration 6 f 6 infl Contors of x,t) Figre. Upper left: Operating chart as Q = Q = m /h. Located on the dashed feed line y = QAf are the feed points of Example, the crosses, and Example, circles. The filled dot is the initial feed point f, s ) =., 7.). Upper right and lower row: Example. A nmerical simlation withot a reglator) when the feed concentration is piecewise constant and periodic with the alternating vales. and. kg/m. Qf t) = m /h, Q t) = Q = L f, s ) = m /h, t) = =. kg/m and e t) =. high vale of the feed flx s implies a jmp from Q ) = 6 when Qmin ) = ) p to min Q ) = Q ) = 6. The amplitdes of Q t), mt), t) and xsb t) are now smaller Feed concentration Contors of x,t) 6 Efflent concentration.. Figre 9. Example. A simlation withot a reglator) where the alternating vales of the periodic feed concentration are and kg/m. This larger amplitde than in Example implies overflow and a slightly declining SBL and mass, on an average. Qf t) =, f, s ) =., 7.), Q t) = Q = L f, s ) = and t) = =.. SDJEM.tex; //7; :; p.

23 A Reglator for Continos Sedimentation Q t) Contors of x,t) Underflow concentration Figre. Example, case A. A simlation sing the same initial data and alternating feed concentration as in = 9, and the alternating vales of Qr t) are Figre. The reglator 6) is applied with K =, Qmax = Q min L.,.) = 6 and L., 9.6) =, respectively. The alternating vales of Qmin = t) are Q and L., 9.6) = 6, respectively, see the dashed lines. COSBL is satisfied. Q t) Contors of x,t) Underflow concentration Figre. Example, case B. A simlation sing the same initial data and alternating feed concentration as in Figre. The reglator 6) is applied with K = and the constant vale Qr = Q =. The alternating min vales of Qmin = and 6, see the dashed lines. COSBL is satisfied. t) are Q than in Figre. Note that the constant Qr implies the following phenomenon. Since the jmp down from s = 7. to st) =. for < t < ) implies that mt) < mr, the reglator decreases Q t) from Q =. After a short while it converges to L.,.) = 6, which is precisely the vale that corresponds to a steady-state soltion in optimal operation with eqal mass flx in and ot), cf. Theorem.. The analogos behavior occrs dring the periods when the feed flx takes the high vale st) = 9.6 < t <, etc.). Then Q t) increases and converges to L., 9.6) =. In accordance with ), the mass also converges to a constant vale, different from mr t), after each jmp. Example, case C. A more slggish behavior of Q t) can be obtained by decreasing the reglator gain K, see Figre. In comparison to Figre where K = ), the vale K =. SDJEM.tex; //7; :; p.

24 S. Diehl Q t) Contors of x,t) Underflow concentration Figre. Example, case C. A simlation sing the same conditions as in Figre, bt with K =. instead. COSBL is satisfied. implies that Q t) and t) are continos and show a more slggish behavior to the price of increased amplitdes in mt) and xsb t). Example, case D. Consider now the control objective COSBL. The initial nderflow concentration is =. and belongs to the optimal-operation steady-state soltion for f, s ), which is the mean vale of the periodically varying feed point. Setting min =. wold ths be a hard constraint to flfil. This corresponds to the rather low maximal bond Qmax =, cf. Figre. Nevertheless, the simlation in Figre shows that COSBL is satisfied. Q t) Contors of x,t) Underflow concentration 6 Figre. Example, case D. A simlation sing the same conditions as in Figre, bt with the additional reqirement t) min =., which is implied by Q t) Qmax =. COSBL is satisfied. Example, case A. Despite the large jmps in the periodic inpt see the circles in Figre ) the reglator can handle the sitation when K =, see Figre. SDJEM.tex; //7; :; p.

25 A Reglator for Continos Sedimentation Q t) Contors of x,t) Underflow concentration Figre. Example, case A. A simlation sing the same initial data and alternating f t) as in Figre 9. The min reglator 6) is applied with K = and the alternating vales of Qmin = and L t) are Q, ) =. = 9. Both these two latter vales are assmed by the reglator, as well as the pper bond Qmax t) = Q Althogh the reglator satrates abot half the time COSBL is satisfied. Example, case B C. Setting Qr t) = Q =, which is the vale corresponding to a steady-state soltion in optimal operation for the mean vale of the two inpt feed points, we get the soltion shown in Figre for K = and Figre 6 for K =.. Q t) Contors of x,t) Underflow concentration Figre. Example, case B. A simlation sing the same conditions as in Figre, bt with the constant mean vale Qr t) = Q = instead. Dring the whole intervals of high load < t <, 6 < t <, etc.) the reglator satrates to Q t) = Qmin t) = L, ) =. Dring the other intervals, however, convergence to Q t) = L, ) = 7 occrs. Althogh the average mass over a period decreases initially, longer simlation times show that it converges to a constant vale and that COSBL is satisfied. Example, case D. Sppose that COSBL is reqired with min =. as in Example D. The constraint Q t) Qmax = implies that there will be particles in the clarification zone dring the high load intervals, since the minimm bond L, ) = > Qmax. A simlation is shown in Figre 7. There will be an overflow with efflent concentrations similar to Figre 9, where Q t) = for t > ), which is close to Qmax =. In SDJEM.tex; //7; :; p.

26 6 S. Diehl Q t) Contors of x,t) Underflow concentration Q t) 6 6 Figre 6. Example, case C. A simlation sing the same conditions as in Figre, bt with K =. instead. The behavior of Q t) is now more slggish and the average mass decreases initially, however, simlation longer see the graphs in the third row) reveals that it converges to a vale sch that the SBL toches the bottom, althogh the average SBL lies within the thickening zone. Figre 9, there is no reglator connected and the nderflow concentration is constant t) =.. However, the mass decreases and the SBL reaches the bottom. The major improvement with the reglator connected is that the SBL is maintained within the thickening zone, which can be confirmed by longer simlation times. 6. Conclding discssions The main reslt in this paper is the nonlinear reglator 6), see the closed-loop system in Figre 6. It consists of a proportional reglator and satrating bonds. The nderlying ideas and reslts originate from the preceding series of papers [ ]. The control objectives we have focsed on are to maintain optimal operation and keep the SBL at a prescribed level, with and withot a constraint on the nderflow concentration, see Section.. A necessary condition for maintaining optimal operation dring long time of dynamic operation is that the feed point satistfies ): f t), st) P Λ Λa. We have in [] motivated why it is reasonable to assme this. If it is not satisfied, the feed concentration is either too high, or the settler is nderdimensioned. Then Q has to be increased sfficiently ) to prevent overflow; see [], where the control of step responses cover all cases. above Q The responses of the process to small distrbances from optimal operation can be fond in Section.. They constitte a fndamental property of the process, which is well known among all operators of clarifier-thickeners: an increase in the control variable will reslt in SDJEM.tex; //7; :; p.6

27 A Reglator for Continos Sedimentation 7 Contors of x,t) Q t) 9 Underflow concentration. Efflent concentration. Figre 7. Example, case D. A simlation sing the same conditions as in Figre, bt with the additional reqirement t) min =., which is implied by Q t) Q max =. This reqirement means that the interval of the satrating bond is redced to the single vale dring the intervals of high load. The settler overflows periodically at the end of these intervals. a decrease in mass and a declining SBL, and vice versa. Note that this conclsion cannot be drawn from the explicit formla ), since t) increases as Q t) decreases, and vice versa. This property, together with the steady-state relation between the mass, the SBL and the control variable in Section., yields the first part of a control strategy, see Section.. This part is realized by means of the proportional reglator 9), which controls the mass in the settler. The key idea is the following. Under dynamic conditions when the settler is in optimal operation, the soltion is approximately like one in optimal operation in steady state. For the latter soltion there is a known relation ) between the mass, the SBL and the control variable. Hence, by controlling the settler sch that optimal operation is maintained, the SBL can be controlled indirectly via the mass. The mass in the settler can be compted since we assme that the inlet and otlet concentrations and volme flows can be measred. In this way, the SBL can be controlled withot measring it. Frthermore, controlling the mass may be more advantageos, since the SBL may vary dring a transient despite the mass is constant. A favorable property of the relationship ) between the mass, SBL and control variable is the following. For a constant control variable, ) is an affine relationship between the mass and the SBL. When the reference vale of the SBL is chosen to be in the middle of the thickening zone, ) is almost constant as a fnction of the control variable, see Figre 9. All nmerical simlations performed by the athor and fond in the literatre converge to steady-state soltions when the feed inpts and the control variable are held constant. The same seems to be tre when the reglator 9) is connected and we conjectre that this is tre. SDJEM.tex; //7; :; p.7