A regulator for continuous sedimentation in ideal clarifier-thickener units. (English) Zbl 1133.76045

Summary: The purpose is to present a regulator for control of continuous-sedimentation process in a clarifier-thickener unit when this is modelled in one space dimension and when the settling properties of the solids obey Kynch’s assumption. The model is a scalar hyperbolic conservation law with space-discontinuous flux function and point source. The most desired type of solution contains a large discontinuity. A common objective is to control the movement of this discontinuity subject to the requirement that the effluent of the process have zero concentration of particles. In addition, there may be a requirement that the underflow concentration of the thickened suspension lie above a predefined value. Based on previous results on the nonlinear behaviour of the process [the author, SIAM J. Appl. Math. 56, No. 2, 388–419 (1996; Zbl 0849.35142)], a nonlinear regulator is presented. It controls the location of the large discontinuity indirectly by controlling the total mass. The process is stabilized significantly and large input oscillations can be handled.


76N25 Flow control and optimization for compressible fluids and gas dynamics
76T20 Suspensions


Zbl 0849.35142
Full Text: DOI


[1] Diehl S (2001). Operating charts for continuous sedimentation I: control of steady states. J Eng Math 41: 117–144 · Zbl 1128.76370
[2] Diehl S (2005). Operating charts for continuous sedimentation II: step responses. J Eng Math 53: 139–185 · Zbl 1086.76069
[3] Diehl S (2006). Operating charts for continuous sedimentation III: control of step inputs. J Eng Math 54: 225–259 · Zbl 1189.76667
[4] Diehl S (2007) Operating charts for continuous sedimentation IV: limitations for control of dynamic behaviour. J Eng Math. doi: 10.1007/s10665-007-9161-7 · Zbl 1154.76047
[5] Diehl S (1995). On scalar conservation laws with point source and discontinuous flux function. SIAM J Math Anal 26(6): 1425–1451 · Zbl 0852.35094
[6] Diehl S (1996). A conservation law with point source and discontinuous flux function modelling continuous sedimentation. SIAM J Appl Math 56(2): 388–419 · Zbl 0849.35142
[7] Bürger R, Karlsen KH, Klingenberg C and Risebro NH (2003). A front tracking approach to a model of continuous sedimentation in ideal clarifier-thickener units. Nonl Anal Real World Appl 4: 457–481 · Zbl 1013.35052
[8] Bürger R, Karlsen KH, Risebro NH and Towers JD (2004). Well-posedness in BV t and convergence of a difference scheme for continuous sedimentation in ideal clarifier-thickener units. Numerische Mathematik 97: 25–65 · Zbl 1053.76047
[9] Karlsen KH and Towers JD (2004). Convergence of the Lax-Friedrichs scheme and stability for conservation laws with a discontinuous space-time dependent flux. Chinese Ann Math Ser B 25(3): 287–318 · Zbl 1112.65085
[10] Kynch GJ (1952). A theory of sedimentation. Trans Faraday Soc 48: 166–176
[11] Bürger R, Karlsen KH and Towers JD (2005). A model of continuous sedimentation of flocculated suspensions in clarifier-thickener units. SIAM J Appl Math 65: 882–940 · Zbl 1089.76061
[12] Bürger R, Karlsen KH and Towers JD (2005). Mathematical model and numerical simulation of the dynamics of flocculated suspensions in clarifier-thickeners. Chem Eng J 111: 119–134
[13] Karlsen KH, Risebro NH and Towers JD (2002). Upwind difference approximations for degenerate parabolic convection-diffusion equations with a discontinuous coefficient. IMA J Numer Anal 22: 623–664 · Zbl 1014.65073
[14] Karlsen KH, Risebro NH, Towers JD (2003) L 1 stability for entropy solutions of nonlinear degenerate parabolic convection-diffusion equations with discontinuous coefficients. Skr. K. Nor. Vidensk. Selsk. 3:49 · Zbl 1036.35104
[15] Stehfest H (1984). An operational dynamic model of the final clarifier. Trans Inst Meas Control 6(3): 160–164
[16] Diehl S, Sparr G, Olsson G (1990) Analytical and numerical description of the settling process in the activated sludge operation. In Briggs R (ed) Instrumentation, control and automation of water and wastewater treatment and transport systems. IAWPRC, Pergamon Press, pp 471–478
[17] Bustos MC, Paiva F and Wendland W (1990). Control of continuous sedimentation as an initial and boundary value problem. Math Methods Appl Sci 12: 533–548 · Zbl 0717.35052
[18] Barton NG, Li C-H and Spencer J (1992). Control of a surface of discontinuity in continuous thickeners. J Austral Math Soc Ser B 33: 269–289 · Zbl 0758.35009
[19] Chancelier J-Ph, Cohende Lara M and Pacard F (1994). Analysis of a conservation PDE with discontinuous flux: a model of settler. SIAM J Appl Math 54(4): 954–995 · Zbl 0811.35077
[20] Chancelier J-Ph, Cohende Lara M, Joannis C and Pacard F (1997). New insight in dynamic modelling of a secondary settler – II. Dynamical analysis. Wat Res 31(8): 1857–1866
[21] Diehl S (1997). Dynamic and steady-state behaviour of continuous sedimentation. SIAM J Appl Math 57(4): 991–1018 · Zbl 0889.35062
[22] Gimse T, Risebro NH (1990) Riemann problems with a discontinuous flux function. In: Engquist B, Gustavsson B (eds) Third international conference on hyperbolic problems, theory, numerical methods and applications, vol I, pp 488–502 · Zbl 0789.35102
[23] Diehl S (2000). On boundary conditions and solutions for ideal clarifier-thickener units. Chem Eng J 80: 119–133
[24] Attir U and Denn MM (1978). Dynamics and control of the activated sludge wastewater process. AIChE J 24: 693–698
[25] Keinath TM (1985). Operational dynamics and control of secondary clarifiers. J Water Pollut Control Fed 57(7): 770–776
[26] Thompson D, Chapmand DT and Murphy KL (1989). Step feed control to minimize solids loss during storm flows. Res J Water Pollut Control Fed 61(11–12): 1658–1665
[27] Johnston RRM and Simic K (1991). Improving thickener operation and control by means of a validated model. Minerals Eng 4: 695–705
[28] Couillard D and Zhu S (1992). Control strategy for the activated sludge process under shock loading. Wat Res 26: 649–655
[29] Kabouris JC, Georgakakos AP and Camara A (1992). Optimal control of the activated sludge process: Effect of sludge storage. Wat Res 26: 507–517
[30] Balslev P, Nickelsen C and Lynggaard-Jensen A (1994). On-line flux-theory based control of secondary clarifiers. Wat Sci Tech 30(2): 209–218
[31] Grijspeerdt K and Verstraete W (1996). A sensor for the secondary clarifier based on image analysis. Wat Sci Tech 33(1): 61–70
[32] Bergh S-G and Olsson G (1996). Knowledge based diagnosis of solids-liquid separation problems. Wat Sci Tech 33: 219–226
[33] Kim Y and Pipes WO (1996). Solids routing in an activated sludge process during hydraulic overloads. Wat Sci Tech 34(3–4): 9–16
[34] Nyberg U, Andersson B and Aspegren H (1996). Real time control for minimizing effluent concentrations during storm water events. Wat Sci Tech 34: 127–134
[35] Sidrak YL (1997). Control of the thickener operation in alumina production. Control Eng Pract 5: 1417–1426
[36] Kim Y, Pipes WO and Chung P-G (1998). Control of activated sludge bulking by operating clarifiers in a series. Wat Sci Tech 38(8–9): 1–8
[37] Dauphin S, Joannis C, Deguin A, Bridoux G, Ruban G and Aumond M (1998). Influent flow control to increase the pollution load treated during rainy periods. Wat Sci Tech 37(12): 131–139
[38] Georgieva PG and FeyoDe Azevedo S (1999). Robust control design of an activated sludge process. Int J Robust Nonlinear Control 9: 949–967 · Zbl 0939.93520
[39] Schoenbrunn F, Toronto T (1999) Advanced thickener control. In: Advanced process control applications for industry workshop. IEEE Industry Applications Society, pp 83–86
[40] Vanderhasselt A, De Clercq B, Vanderhaegen B, Vanrolleghem P and Verstraete W (1999). On-line control of polymer addition to prevent massive sludge washout. J Env Eng 125: 1014–1021
[41] Charef A, Ghauch A and Martin-Bouyer M (2000). An adaptive and predictive control strategy for an activated sludge process. Bioprocess Biosystems Eng 23: 529–534
[42] Georgieva P and Ilchmann A (2001). Adaptive lambda-tracking control of activated sludge processes. Int J Control 12: 1247–1259 · Zbl 1027.93028
[43] Ito H (2004) A dissipative approach to control of biological wastewater treatment plants based on entire nonlinear process models. In: Proceedings of the 2004 American control conference, vol 6. IEEE, pp 5489–5495
[44] Wilén B-M, Lumley D and Nordqvist A (2004). Dynamics in maximal settling capacity in an activated sludge treatment plant with highly loaded secondary settlers. Wat Sci Tech 50: 187–194
[45] Olsson G, Nielsen MK, Yuan Z, Lynggaard-Jensen A, Steyer J-P (2005) Instrumentation, control and automation in wastewater systems. Intl Water Assn
[46] Jeppsson U, Rosen C, Alex J, Copp J, Gernaey KV, Pons M-N and Vanrolleghem PA (2006). Towards a benchmark simulation model for plant-wide control strategy performance evaluation of WWTPs. Wat Sci Tech 53: 287–295
[47] Diehl S and Jeppsson U (1998). A model of the settler coupled to the biological reactor. Wat Res 32(2): 331–342
[48] Maljian MV and Howell JA (1978). Dynamic response of a continuous thickener to overloading and underloading. Trans Ind Chem Eng 56: 55–61
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