×

zbMATH — the first resource for mathematics

Operating charts for continuous sedimentation. II: Step responses. (English) Zbl 1086.76069
Summary: [For part I see the author, ibid. 41, No. 2–3, 117–144 (2001; Zbl 1128.76370).]
The process of continuous sedimentation of particles in a liquid has often been predicted by means of operating charts and mass-balance considerations, where the underlying constitutive assumption is the one by G. J. Kynch [Trans. Faraday Soc. 48, 166–176 (1952)]. Much more complex operating charts (concentration-flux diagrams) can be obtained from a one-dimensional model of an ideal continuous clarifier-thickener unit. The engineering concept of ’optimal operation’ is defined generally as a special type of solution of the model equation, which is a conservation law with a source term and a space-discontinuous flux function. All qualitatively different step responses (with the unit initially in optimal operation in steady state) are presented and classified in terms of operating charts. Quantitative information relating several interesting variables are also presented concerning, for example, the time until overflow occurs as a function of the feed concentration and flux.

MSC:
76T20 Suspensions
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] S. Diehl, Operating charts for continuous sedimentation I: Control of steady states. J. Engng. Math. 41 (2001) 117-144 · Zbl 1128.76370 · doi:10.1023/A:1011959425670
[2] G.J. Kynch, A theory of sedimentation. Trans. Faraday Soc 48 (1952) 166-176 · Zbl 0048.22902 · doi:10.1039/tf9524800166
[3] ??. Jernqvist, Experimental and theoretical studies of thickeners. Part 2. Graphical calculation of thickener capacity. Svensk Papperstidning 68 (1965) 545-548
[4] ??. Jernqvist, Experimental and theoretical studies of thickeners. Part 3. Concentration distribution of the steady and unsteady state operation of thickeners. Svensk Papperstidning 68 (1965) 578-582
[5] T.M. Keinath, M. Ryckman, C. Dana and D. Hofer, Activated sludge ??? unified system design and operation. J. Envir. Engng Div., ASCE 103 (1977) 829-849
[6] V.D. Laquidara and T.M. Keinath, Mechanism of clarification failure. J. Water Pollut. Control Fed. 55 (1983) 54-57
[7] H. Stehfest, An operational dynamic model of the final clarifier. Trans. Inst. Meas. Control 6 (1984) 160-164 · doi:10.1177/014233128400600307
[8] T.M. Keinath, Operational dynamics and control of secondary clarifiers. J. Water Pollut. Control Fed. 57 (1985) 770-776
[9] B.F. Severin, Clarifier sludge-blanket behaviour at industrial activated-sludge plant. J. Environ. Engng. 117 (1991) 718-730 · doi:10.1061/(ASCE)0733-9372(1991)117:6(718)
[10] N.G. Barton, C.-H. Li and J. Spencer, Control of a surface of discontinuity in continuous thickeners. J. Austral. Math Soc. Ser. B 33 (1992) 269-289 · Zbl 0758.35009 · doi:10.1017/S0334270000007050
[11] R. B??rger and W.L. Wendland, Sedimentation and suspension flows: Historical perspective and some recent developments. J. Engng. Math. 41 (2001) 101-116 · Zbl 1014.76003 · doi:10.1023/A:1011934726111
[12] S. Hasselblad and S.L. Xu, Solids separation parameters for secondary clarifiers. Water Environ. Res. 70 (1998) 1290-1294 · doi:10.2175/106143098X123660
[13] C.M. Bye and P.L. Dold, Evaluation of correlations for zone settling velocity parameters based on sludge volume index-type measures and consequences in settling tank design. Water Environ. Res. 71 (1999) 1333-1344 · doi:10.2175/106143096X122348
[14] W.T. Manning, M.T. Garrett and J.F. Malina, Sludge blanket response to storm surge in an activated-sludge clarifier. Water Environ. Res. 71 (1999) 432-442 · doi:10.2175/106143097X122059
[15] G. Kaushik and Z.V.P. Murthy, Thickener design: The new method. Chemical Engineering World 37 (2002) 145-147
[16] B. Wett, A straight interpretation of the solids flux theory for a three-layer sedimentation model. Wat. Res. 36 (2002) 2949-2958 · doi:10.1016/S0043-1354(01)00523-1
[17] G.B. Wallis, One-dimensional Two-phase Flow. New York: McGraw-Hill (1969). · Zbl 0174.29503
[18] C.A. Petty, Continuous sedimentation of a suspension with a nonconvex flux law. Chem. Engng. Sci. 30 (1975) 1451-1458 · doi:10.1016/0009-2509(75)85022-6
[19] H.K. Rhee, R. Aris and N. Amundson, First-Order Partial Differential Equations, volume 1. Englewood Cliffs: Prentice-Hall (1986). · Zbl 0699.35001
[20] M.C. Bustos and F. Concha, On the construction of global weak solutions in the Kynch theory of sedimentation. Math Methods Appl. Sci. 10 (1988) 245-264 · Zbl 0679.35063 · doi:10.1002/mma.1670100304
[21] M.C. Bustos, F. Concha and W. Wendland, Global weak solutions to the problem of continuous sedimentation of an ideal suspension Math. Methods Appl. Sci. 13 (1990) 1-22 · Zbl 0722.35054 · doi:10.1002/mma.1670130102
[22] M.C. Bustos, F. Paiva and W. Wendland, Control of continuous sedimentation as an initial and boundary value problem. Math Methods Appl. Sci. 12 (1990) 533-548 · Zbl 0717.35052 · doi:10.1002/mma.1670120607
[23] S. Diehl, G. Sparr and G. Olsson, Analytical and numerical description of the settling process in the activated sludge operation. In: R. Briggs (ed.), Instrumentation, Control and Automation of Water and Wastewater Treatment and Transport Systems. IAWPRC, Pergamon Press (1990) pp. 471???478.
[24] T. Gimse and N. H. Risebro, Riemann problems with a discontinuous flux function. In: B. Engquist and B. Gustavsson (eds.), Third International Conference on Hyperbolic Problems, Theory, Numerical Methods and Applications, volume I (1990) pages pp 488???502. · Zbl 0789.35102
[25] S. Diehl, On scalar conservation laws with point source and discontinuous flux function. SIAM J. Math. Anal. 26 (1995) 1425-1451 · Zbl 0852.35094 · doi:10.1137/S0036141093242533
[26] S. Diehl, A conservation law with point source and discontinuous flux function modelling continuous sedimentation. SIAM J Appl. Math. 56 (1996) 388-419 · Zbl 0849.35142 · doi:10.1137/S0036139994242425
[27] S. Diehl, On boundary conditions and solutions for ideal clarifier-thickener units. Chem. Engng. J. 80 (2000) 119-133 · doi:10.1016/S1383-5866(00)00081-2
[28] J.-Ph. Chancelier, M. Cohende Lara and F. Pacard, Analysis of a conservation PDE with discontinuous flux: A model of settler SIAM J. Appl. Math. 54 (1994) 954-995 · Zbl 0811.35077
[29] J.-Ph. Chancelier, M. Cohende Lara, C. Joannis and F. Pacard, New insight in dynamic modelling of a secondary settler ??? I. Flux theory and steady-states analysis. Wat. Res. 31 (1997) 1847-1856 · doi:10.1016/S0043-1354(96)00286-2
[30] J.-Ph. Chancelier, M. Cohende Lara, C. Joannis and F. Pacard, New insight in dynamic modelling of a secondary settler ??? II Dynamical analysis. Wat. Res. 31 (1997) 1857-1866 · doi:10.1016/S0043-1354(96)00287-4
[31] R. B??rger, K.H. Karlsen, C. Klingenberg and N.H. Risebro, A front tracking approach to a model of continuous sedimentation in ideal clarifier-thickener units. Nonl. Anal. Real World Appl. 4 (2003) 457-481 · Zbl 1013.35052 · doi:10.1016/S1468-1218(02)00071-8
[32] R. B??rger, K.H. Karlsen, N.H. Risebro and J.D. Towers, Well-posedness in BV t and convergence of a difference scheme for continuous sedimentation in ideal clarifier-thickener units. Numerische Mathematik 97 (2004) 25-65 · Zbl 1053.76047 · doi:10.1007/s00211-003-0503-8
[33] K.H. Karlsen and J.D. Towers, Convergence of the lax-friedrichs scheme and stability for conservation laws with a discontinuous space-time dependent flux. Chinese Ann. Math. Ser. B 25 (2004) 287-318 · Zbl 1112.65085 · doi:10.1142/S0252959904000299
[34] P. Garrido, R. Burgos, F. Concha and R. B??rger, Software for the design and simulation of gravity thickeners. Miner Engng. 16 (2003) 85-92 · doi:10.1016/S0892-6875(02)00168-1
[35] A. Zeidan, S. Rohani and Z. Bassi, BioSys: Software for wastewater treatment simulation. Adv. Engng. Software 34 (2003) 539-549 · Zbl 02024065 · doi:10.1016/S0965-9978(03)00072-3
[36] R. B??rger, K.H. Karlsen, N.H. Risebro and J.D. Towers, On a model for continuous sedimentation in vessels with discontinuously varying cross-sectional area. In: T.Y. Hou and E. Tadmor (eds.) Hyperbolic Problems: Theory, Numerics, Applications Proceedings of the Ninth International Conference on Hyperbolic Problems Held in CalTech, Pasadena. March 25-29, 2002. Berlin: Springer Verlag (2003) pp. 397-406 · Zbl 1059.76542
[37] R. B??rger, J.J.R. Damasceno and K.H. Karlsen, A mathematical model for batch and continuous thickening of flocculated suspensions in vessels with varying cross-section Int. J. Miner. Process. 73 (2004) 183-208 · doi:10.1016/S0301-7516(03)00073-5
[38] R. B??rger, K.H. Karlsen, N.H. Risebro and J.D. Towers, Numerical methods for the simulation of continuous sedimentation in ideal clarifier-thickener units. Int. J. Mineral Process 73 (2004) 209-228 · doi:10.1016/S0301-7516(03)00074-7
[39] R. B??rger, K.H. Karlsen, N.H. Risebro and J.D. Towers, Monotone difference approximations for the simulation of clarifier-thickener units. Comput. Vis. Sci. 6 (2004) 83-91 · Zbl 1299.76283
[40] S. Berres, R. B??rger and K.H. Karlsen, Central schemes and systems of conservation laws with discontinuous coefficients modeling gravity separation of polydisperse suspensions. J Comp. Appl. Math. 164???165 (2004) 53-80 · Zbl 1107.76366 · doi:10.1016/S0377-0427(03)00496-5
[41] R. B??rger, K. H. Karlsen and N. H. Risebro, A relaxation scheme for continuous sedimentation in ideal clarifier-thickener units. To appear in Computers Math. Appl. · Zbl 1122.76063
[42] R. B??rger, K.H. Karlsen and J.D. Towers, A model of continuous sedimentation of flocculated suspensions in clarifier-thickener units. SIAM. J. Appl. Math. 65 (2005) 882-940 · Zbl 1089.76061 · doi:10.1137/04060620X
[43] K.H. Karlsen, N.H. Risebro and J.D. Towers, Upwind difference approximations for degenerate parabolic convection-diffusion equations with a discontinuous coefficient. IMA J. Numer Anal. 22 (2002) 623-664 · Zbl 1014.65073 · doi:10.1093/imanum/22.4.623
[44] K. H. Karlsen, N. H. Risebro and J. D. Towers, L 1 stability for entropy solutions of nonlinear degenerate parabolic convection-diffusion equations with discontinuous coefficients Skr. K. Nor. Vidensk. Selsk. 3 (2003) 49??pp. · Zbl 1036.35104
[45] S. Diehl, Scalar conservation laws with discontinuous flux function: I. The viscous profile condition. Comm Math. Phys. 176 (1996) 23-44 · Zbl 0845.35067 · doi:10.1007/BF02099361
[46] S. Diehl and N.-O. Wallin, Scalar conservation laws with discontinuous flux function: II. On the stability of the viscous profiles. Comm. Math. Phys. 176 (1996) 45-71 · Zbl 0845.35068 · doi:10.1007/BF02099362
[47] S. Diehl and U. Jeppsson, A model of the settler coupled to the biological reactor. Wat. Res. 32 (1998) 331-342 · doi:10.1016/S0043-1354(97)00048-1
[48] M.V. Maljian and J.A. Howell, Dynamic response of a continuous thickener to overloading and underloading. Trans. Ind. Chem Engng. 56 (1978) 55-61
[49] Oleinik O.A., Uniqueness and stability of the generalized solution of the Cauchy problem for a quasi-linear equation. Uspekhi Mat. Nauk. 14 (1959) 165???170. Amer. Math. Soc Trans. Ser. 2 33 (1964) 285???290.
[50] D.P. Ballou, Solutions to nonlinear hyperbolic Cauchy problems without convexity conditions. Trans. Amer. Math. Soc. 152 (1970) 441-460 · Zbl 0207.40401 · doi:10.1090/S0002-9947-1970-0435615-3
[51] S. Diehl, Operating charts for continuous sedimentation III: Control of step inputs. To appear in J. Engng. Math. (2006). · Zbl 1189.76667
[52] S.K. Godunov, A finite difference method for the numerical computations of discontinuous solutions of the equations of fluid dynamics. Mat. Sb. 47 (1959) 271-306 · Zbl 0171.46204
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.