athens, iroon polytechniou 5, 15773 zografou, athens, greece · invited paper modelling of settling...

Invited Paper

Modelling of settling tanks - a critical review

A.I. Stamou

Department of Civil Engineering, National Technical University of

Athens, Iroon Polytechniou 5, 15773 Zografou, Athens, Greece

Abstract

Modelling of settling tanks started in 1905, when Hazen presented his simpleone-dimensional laminar flow model with single-sized solid particles. In theperiod 1904-1967 simple theoretical models were developed, which assumed asimplified (e.g. a uniform) flow pattern. Between 1967 and 1977 empiricalmodels followed, which correlated removal efficiency with important designparameters. Since 1976 advanced, two-dimensional, numerical models have beendeveloped. These model have been applied to settling tanks with increasinggeometrical and process complexity, ranging from simple rectangular tankswithout density effects to tanks with complicated geometries with inletdeflectors, inlet vanes, sludge hoppers and density effects. Future models forsettling tanks are expected to be three-dimensional, computationally moreefficient and including biological parameters as process variables. Thesemodels would be capable to handle complicated tank geometries and deal withproblems encountered in settling tanks, such as sludge rising and bulking.

1 Introduction

Settling by gravity is the most common and extensively applied treatmentprocess for the removal of Suspended Solids (SS) from water and wastewater.Since the investment for settling tanks in treatment plants is high (30% of thetotal investment), the calculation of the SS removal efficiency has been thesubject of numerous theoretical and experimental studies. The removal effi-ciency depends on the characteristics of the SS (e.g. particle size, density,settling velocity) and the flow field (e.g. flow velocities, turbulent eddyviscocity distribution) in the tank.

Determination of the flow field, the SS-concentration field and the removalefficiency can be performed with the use of mathematical models. Thesemodels range from simplified theoretical or empirical expressions to moresophisticated numerical models. In the present paper a concise literature reviewof the most representative mathematical models is presented.

Transactions on Ecology and the Environment vol 7, © 1995 WIT Press, www.witpress.com, ISSN 1743-3541

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2 Existing mathematical models

Simple theoretical modelsIn 1904 Hazen[l] presented his ideal settling model, which assumes a laminar anduniform one-dimensional flow and SS having a single size and a single settlingvelocity. Hazen introduced the term "overflow rate" (OR) and concluded thatthe critical parameter in the design of settling tanks is the surface area, whilethe depth is not of importance.

Forty years later Dobbins[2] presented his model, which is the analyticalsolution of the 1-dimensional advection-diffusion equation for the SS underspecific boundary conditions with a constant turbulent eddy viscocity in thedirection of the depth. Camp[3] modified Hazen's model by introducing aSettling Velocity Curve (SVC) for the SS, instead of a single settling velocity.Both Dobbins[2] and Camp[3] concluded that turbulence delays the settling ofSS and thus reduces the SS removal efficiency. Camp also examined the effectof coagulation between SS particles and concluded that it is due to differencesin horizontal and vertical velocities. Furthermore, he introduced the conceptof the Flow Through Curve (FTC).

In 1956 Fitch[4] introduced his model, which showed that the removalefficiency depends only on the settling velocity of the SS and the OR, while therole of depth is not important. He also noted that provided the distribution offlow in the direction of the width is uniform, density differences and tankgeometry are not important factors. Weidner[5] used some fundamentalhydraulic theories (e.g. flow between parallel plates) to describe thedistribution of the flow velocity in a rectangular settling tank.

The main characteristics of these models are : (i) they all deal withrectangular tanks, (ii) the horizontal flow velocity is assumed to follow a simplepattern (e.g. uniform) and turbulence is either ignored or modeled by a constantvertical eddy viscocity and (iii) the concentration of the SS is determined bythe analytical solution of the one-dimensional, advection-dif fusion equation.

Empirical modelsIn the period 1968-1978 a number of empirical models was presented, whichpredicted the removal efficiency using empirical design parameters. Thesemodels, which were introduced mainly by public health engineers, ignored theflow regime in the tank. Indicatively, the models of Voshel and Sak [6] andTebbutt and Christoulas [7] are mentioned. Voshel and Sak[6] used a multipleregression analysis to derive an equation for the determination of the removalefficiency on the basis of the OR, the solids loading and the use of chemicals.Tebbutt and Christoulas [7] using linear regression derived an empiricalequation for primary settling tanks, which predicted the removal efficiencyusing the OR and the inlet SS concentration.

Two dimensional advanced numerical modelsModels with simplified tank geometry without buoyancy. In 1977Larsen[8] presented a model for rectangular activated sludge clarifiers. He usedthe DuFort-Frankel differential scheme for the discretization of the non-conservative form of the vorticity transport equation and the convection-


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diffusion equation for the SS. Larsen also carried out extensive experimentalstudies in model and real settling tanks, which indicated the level anddistribution of turbulence, as well as 3-dimensional effects.

Larsen's pioneering work was followed by research carried out mainly byresearch groups in the University of Davies in California, the University ofWindsor in Ontario and the University of Karlsruhe in Germany.

Schamber and Larock[9,10] from the University of Davies presented amodel to predict the flow in a rectangular tank, with a simple geometryresembling the "flow over a cavity" problem, i.e. the tank was fed through aninlet channel whose water surface was the same as the water tank surface.Schamber and Larock employed the k-e model of turbulence[ll], which allowsto simulate the non-uniform distribution of the eddy viscocity /diffusivity.They applied the Galerkin finite element method for solving the steady, 2-dimensional Reynolds equations. In their model, a hydrostatic pressuredistribution was assumed without verification. The Newton based solutionalgorithm used in the model was very sensitive to initial values of thedependent variables. No comparison was made to experiments.

In the University of Windsor both experimental and computational workwere carried out. Imam and McCorquodale[12] developed a 2-dimensional,finite-difference model for the simulation of flow in rectangular settling tankswith a somewhat realistic inlet geometry with an inlet baffle. A two-step ADI,3-time level, weighted upwind-centered difference scheme was used to solvethe vorticity-transport equation, while the centered difference analog of thestream function equation was solved using the SOR method. Imam andMcCorquodale[12] carried out flow field experiments to calibrate (i.e. determinethe constant value of the eddy viscocity) and verify the model. Imam et al[14]extended this flow field model by adding a model for the SS. The SS modelconsisted of an unsteady, 2-dimensional convection-diffusion equation withadditional terms accounting for the settling, deposition and scouring of the SS.The settling velocity was assumed to be constant, i.e. the model was restrictedto non-flocculant, mono-dispersed SS. The model was applied to study theeffect of the inlet baffle submergence on the removal of SS. Predicted FTCswere compared to experiments. Abdel-Gawad and McCorquodale [14,15]proposed the strip integral method, a forward marching scheme in which thepartial differential equations of continuity and momentum are reduced to a setof ordinary differential equations in terms of certain preselected parameters.The latter along with a set of shape functions describe the velocity distribution.The shape functions were chosen to allow recirculation behind the inlet baffle.Flow field calculations were compared favourably with experimental flowvelocities performed in a model settling tank. Predicted SS concentrations andremoval efficiencies were compared to experimental values obtained in the realrectangular tank in the city of Sarnia at Ontario.

The first model presented by researchers [16,17,18] of the Institute ofHydromechanics of the University of Karlsruhe used an ADI type finitedifference method to solve the partial differential equations governing thesteady 2-dimensional turbulent flow in the vertical plane of rectangular settlingtanks along with the k-e model of turbulence. The hybrid scheme was used forspatial discretization. The pressure distribution was calculated from an iterativeprocedure based on the "SIMPLE" algorithm. An additional differentialequation governing the unsteady transport of dye was solved using the hybriddiscretization scheme to calculate the FTC. The model simulated satisfactorily


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the experiments of Imam and McCorquodale[12], although FTC computationshave shown somewhat excessive mixing, which was attributed to the numericaldiffusion involved with the numerical scheme. The second model, which waspresented by Stamou et al[191 used also the k-e model of turbulence. The modelconsisted of a flow model providing the velocity and turbulent viscocity/diffusivity field and a SS transport model The SS transport model consistedof a number of SS transport equations, which were solved for individualfractions of different particle sizes having different settling velocities (i.e .aSVC). From the predicted concentration fields for the individual fractions aweighted average concentration field was determined. The main characteristicsof the model were the use of (i) the "SIMPLEC" algorithm for the pressurecorrection equation, (ii) the Stone's strongly implicit solution algorithm insteadof the conventional Line by Line and (iii) the third order accurate QUICKscheme for the spatial discretization of the convection terms for thedetermination of the FTC to reduce significantly numerical diffusion.Predicted flow velocities, FTCs, SS concentrations and removal efficiencieswere compared to experimental values obtained in the real rectangular tank inthe city of Sarnia, Ontario. Adams and Rodi [20] applied the model of Stamouet al[19] to predict satisfactorily the velocity and FTC measurements obtainedin an experimental model, consisting of an inlet slot and an outlet weir. A smalldifference between computations and experiments was attributed to the effectsof flow curvature, which are not included in the standard k-e model.Computations] using a modified version of the k-e model, which accounts forthe effects of mean streamline curvature showed better agreement withmeasurements.

The main characteristics of these models are the following:(i) They deal with tanks having geometry, which is rectangular and simple,

consisting of an inlet slot and an outlet weir. Likewise, the numerical grid isalso simple and orthogonal.

(ii) The flow in the settling tanks is assumed to be steady, two-dimensional(i.e. all parameters are uniformly distributed in the direction of the width, andsingle phase (i.e. only the liquid phase is considered, without density effects dueto the presence of SS). This assumption can be initially considered asconservative for primary settling tanks.

(iii) The SS are assumed to be discrete, single-sized and having a singlesettling velocity or a size distribution described by a SVC. Coagulation istherefore ignored. Since coagulation would enhance sedimentation, the removalefficiencies predicted by models ignoring coagulation can be considered asconservative.

(iv) For the calculation of the flow field the continuity and momentumequations or the vorticity transport and the stream function equations aresolved. The turbulence is described by a single value of the eddy viscocity orwith the aid of the k-e model, in its standard or modified version.

(v) For the calculation of the SS concentration field and the removalefficiency one or more convection-diffusion equations for the SS are solved.The solution of the SS concentration field is decoupled from the flow field.

The scope of these models was to develop verified mathematical modelsto be used as design tools for primary settling tanks Also, experimental studieshave been performed in laboratory tanks with simple geometrical configurationwith pure water to produce data for the verification of the models. It is notedthat the majority of the existing field data from real settling tanks could hardly


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be used for model verification, due mainly to the very complicated three-dimensional inlet geometries [22].

Models with complicated tank geometry and buoyancy. In 1986 Adamsand Stamou[23] modified the model of Stamou et al[191 by adding the buoyancyforce terms in the momentum equations. The model was applied to predict thevelocity and the SS-concentration fields in the tertiary, rectangular settlingtanks at Almhult, Sweden[8] after a significant number of simplifications. The3-dimensional inlet of the tank was approximated by an inlet slot positioned atmid-width and the outlet launders by a series of outlet 2-dimensional pipes.Calculations without buoyancy showed a unidirectional flow throughout thetank after a short separation region near the inlet. Calculations with buoyancyshowed, as in the real secondary settling tanks, the typical "two layer flow",which consisted of a bottom current and a free surface return current,occupying the entire top half of the tank. SS concentration calculations werenot very successful. This was attributed to the simulation of settling with justone settling velocity. Furthermore, computational difficulties were encounteredwith convergence, occurring at regions with strong buoyancy, where turbulenceenergy is driven to zero and the k-e model would no longer function correctly.

In 1987 Devantier and Larock[24] presented a finite-element model forpredicting the sediment induced density currents in circular radial flow tanks.The model employed the continuity and momentum equations and theconservation equation for the SS, along with a modified version of the k-eturbulence model for stratified flow[ll]. A scouring parameter was included inthis model, while coagulation was ignored. The model was applied to a simpletank with an inlet under a baffle,. The typical "two layer flow" was observedwith a strong bottom current and a free surface return current. No comparisonwas made to experiments.

Lyn et al[25] presented a model with a nonorthogonal grid formulation,which could handle fairly arbitrary 2-dimensional geometries and permitted themodelling of the sludge hopper. The model solved the full 2-dimensional ellipticgoverning flow equations with the k-e turbulence model. Effects onmomentum, as well as on k and e due to sediment induced density currentswere taken into account. The SS were treated by considering a SVC. A simplef locculation model, which assumed only turbulent shear-induced f locculationwas incorporated into the model. In the past numerical models of f locculation[26] have assumed idealized spatially homogeneous flows. Predicted velocityand SS-concentration profiles and removal efficiencies were compared withobservations in a tertiary settling tank in Sweden [8,23]. Results suggested thatthe flow field was significantly changed by seemingly small density differences,that shear-induced flocculation was relatively unimportant, and that theinfluent settling velocity distribution was critical for accurately predicting ofremoval efficiencies. Lyn et al[25] noted also the substantially increasedcomputational times, when density effects were included in the model, due toslow convergence (e.g. use of very low underelaxation factors).

Zhou and McCorquodale [27] predicted the "two layer flow" in secondarysettling tanks and verified their model by three field investigations. They useda modified version of the k-e turbulence model, but they ignored the effectsof density in the k and e equations at a first approximation. As observed byStamou [28] the implementation of the large buoyancy term in the k equationmakes this equation numerically stiff and thus special measures should be taken


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to obtain converged solutions, such as the use of extremely low values ofunderelaxation factors, which may delay significantly convergence. Zhou andMcCorquodale[27] observed a high entrainment of ambient fluid into thesinking influent flume, which was reduced by the use of inlet baffles. They alsonoted that the removal efficiency was strongly related to the SScharacteristics, and suggested the use of a settling velocity distribution [29],which takes into account the size non-uniformity of the SS.

Lyn and Zhang[31] used the mathematical model of Lyn et al [25] tosimulate the non-buoyant flow field and the FTC in the fairly complexgeometrical configuration of a centrally fed circular model tank. Szalai etal[32] repeated the calculations of Lyn and Zhang by including the effect ofswirl and by making a number of improvements in the mathematical model.The inclusion of swirl allowed the model to account for the influence of thecircumferential sludge removal mechanism (i.e. a scraper) as well as for theeffect of swirl inducing vanes at the inlet. The main improvements were thevectorization of the code and the use of the HLPA discretization scheme forthe convection terms to reduce numerical diffusion. Computations showedclearly that the radial flow was strongly influenced by swirl. In the case ofswirling flow, a momentum source was added to the radial flow and theproduction of k was increased affecting significantly the flow field.

Zhou et al[30] has recently presented a model, which described the flowregime and the temperature mixing in temperature stratified settling tanksassociated with a warm influent. The model consisted of a series ofconservation equations for fluid mass, momentum and temperature. For thedescription of turbulence the k-e model and an algebraic stress model wereused. Model predictions were compared to laboratory experiments.

The general characteristics of these models are the following:(i) They deal with tanks which have a realistic and thus more complicated

geometry. Thus, the numerical grid is usually non-orthogonal.(ii) The flow in the settling tanks is still assumed to be steady and

two-dimensional or axisymmetric. However, the density effects due to thepresence of SS are taken into account and the solution of the SS concentrationfield is coupled to the flow field..

(iii) The SS are assumed to be non-uniform and coagulation is not ignored.

3 Future perspectives-Discussion

Numerical models for settling tanks have almost reached the stage that theycan be used by design engineers to optimize the design of settling tanks.

Taking into account the extremely fast progress in the computertechnology, the following requirements are expected to be fulfilled in the nearfuture, towards the formulation of perfect mathematical models:

(i) The models should be able to simulate the complicated three-dimensional tank geometries of real tanks. This can be performed with the useof non-orthogonal computational grids.

(ii) The models should be able to cope with strong buoyancy effects,encountered in real secondary settling tanks. For this reason an appropriateturbulence closure scheme (e.g. a modified version of the k-e model) is required,combined with a computationally effective solution method, which would allowa fast calculation of the coupled flow and SS concentration fields.


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(Hi) The models should be able to model the transient behaviour of the SSconcentration field in the case of sudden increase of flowrate. Thus, they shouldbe able to predict the process of SS washout in the event of a rain, everyimportant practical problem. For this reason both flow field and SS transportequations should be written and solved in unsteady formulation.

(iv) The models should include biological parameters as process variables,so as to be able to model very important bio-chemical processes and problemsencountered in real secondary settling tanks, such as sludge rising, sludgebulking and biological phosphorus removal. For this reason, the modelling ofsecondary settling tanks should be combined with models of the biologicalreactors [33,34].

To satisfy these requirements, proper experimental data are required to becollected for model verification. This is expected to be a very difficult andtime-consuming task.

4 Acknowledgements

The financial support of the General Secretariat of Research and Technologyof Greece to the author for participation in the present Conference isacknowledged herewith.

5 References

1. Hazen, A. On Sedimentation, Transactions, ASCE, 1904, 53, 45-71.2. Dobbins,W.R Effects of turbulence on sedimentation, Transactions, ASCE,

1944,109, 629-656.3. Camp, T. R. Sedimentation and the design of settling tanks, 1946,

Transactions, ASCE, 111, 895-936.4. Fitch,E.B. Biological treatment of sewage and industrial wastes,

Sedimentation process fundamentals, ed. McCabe and Eckenfelder,Reinhold Publishing Corporation, 1958.

5. Weidner, J. Zufluss, Durchfluss und Absetzwirkung ZweckmaessigGestalteter Rechtbecken, Kommissionsverlag R.Oldenbourg, Muenchen,1967.

6. Voshel, D. and Sak, J. G. Effect of primary effluent suspended solids andBOD on activated sludge production, 1968, J.of WPCF, 40, 203-212.

7. Tebbutt, T. H. and Christoulas, D. G. Performance relationships for primarysedimentation, 1975, Water Research, 9, 347-356.

8. Larsen, P. On the hydraulics of rectangular settling basins-Experimentaland theoretical studies, Report No.1001, Dept. of Water ResourcesEngineering, Lund Institute of Technology, Lund, Sweden, 1977.

9. Schamber, D. R. and Larock, B. E. A Finite element model of turbulentflow in primary sedimentation basins, pp.3.3 to 3.21, Proceedings on FiniteElement in Water Resources, FE2, London, England, 1978.

10. Schamber, D. R. and Larock, B. E. Numerical analysis of flow in sedimen-tation basins, 1981, / of' Hydr. Div., ASCE, 107, 575-591.

11. Rodi, W. Turbulence Models and Their Application in Hydraulics -A Stateof the Art Review, IAHR, Delft, The Netherlands, 1980.

12. Imam, E. and McCorquodale, J. A. Simulation of flow in rectangularclarifiers, 1983, J.of Env.Engrg., ASCE, 109, 713-730.

13. Imam, E., McCorquodale, J. A. and Bewtra, J. K. Numerical modelling ofsedimentation tanks, 1983, J.of Hydr.Engrg, ASCE, 109, 1740-1754.


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14. Abdel-Gawad, S. M. and McCorquodale, J.A. Strip integral method appliedto settling tanks, 1984, J.of Hydr.Engrg., ASCE, 110, 1-17.

15. Abdel-Gawad, S. M. and McCorquodale, J. A. Numerical simulation ofrectangular settling tanks, 1985, J.of IAHR, 23, 85-100.

16. Celik, L and Rodi, W. Simulation of hydrodynamics and transport characte-ristics of rectangular settling tanks, Euromech 192, Transport of SuspendedSolids in Open Channels, pp.129 to 132, Neubiberg, Germany,1985.

17. Celik, I, Rodi, W. and Stamou, A. I. Prediction of hydrodynamic characte-ristics of rectangular settling tanks, Turbulence Measurements and FlowModelling, ASCE, New York, N.Y., 1985

18. Stamou, A. I. and Adams, E. W. Description of the VEST computer codewith examples of applications to settling tanks, Report SFB 210/T/16,SFB210, University of Karlsruhe, Germany, 1985.

19. Stamou, A. L, Adams, E. W. and Rodi, W. Numerical simulation of flow andsettling in primary rectangular clarifiers, 1989, / of IAHR, 27, 665-682.

20. Adams, E. W. and Rodi, W. Modelling flow and mixing in sedimentationtanks, 1990, J.of'Hydr.Engrg., ASCE, 116, 895-913.

21. Stamou, A. L On the Prediction of flow and mixing in settling tanks usinga curvature modified k-e model, 1991, / of Appl. Math. Model.., 15, 351-358.

22. Stamou, A. L and Rodi, W. Review of experimental studies onsedimentation tanks, Report SFB 210/E/2, SFB210, University of Karlsruhe,Germany, 1984.

23. Adams, E. W. and Stamou, A. I. A computational evaluation of buoyancyeffects in settling tanks, Proc. of Int. Symposium on Buoyant Flows, NTUA,Athens, Greece, pp.418 to 430, 1986.

24. DeVantier, B. A. and Larock, B. E. Modelling sediment-induced densitycurrents in sedimentation basins, 1987, J.of Hydr.Eng., ASCE, 113, 80-94.

25. Lyn, D. A., Stamou, A. I. and Rodi, W. Density currents and shear inducedflocculation in sedimentation tanks, 1992, J.of Hydr.Eng ASCE, 118, 849-867.

26. Valioulis, I. A. and List, E. J. Numerical simulation of a sedimentation basin,1984, /o/#oK .%%eace aW TbaWA 18, 242-253.

27. Zhu, S., and McCorquodale, J. A. Modelling of rectangular settling tanks,1992, / o// G awyc#7#, ASCE, 118, 1391-1405.

28. Stamou, A. Discussion of Modelling of rectangular settling tanks, 1994, /o/#r raw//c#% ASCE, 120, 277-279.

29. Tacacs, L, Patry, G. and Nolasco, D. Dynamic model of clarifier-thicheningprocess, 1991, 7o/ tKaferTfejearc/r, 25,1263-1271.

30. Zhou, S., McCorquodale, J. A. and Godo, A. M. Short circuiting and densityinterface in primary clarifiers, 1994, / of Hydraulic Eng., ASCE, 120, 1060-1080.

31. Lyn, D. A. and Zhang? Z. Boundary fitted numerical modelling of sedimen-tation tanks, Proc. 23.IAHR Congress, A331-A338, Madrid, Spain, 1989.

32. Szalai, L., Krebs, P. and Rodi, W. Simulation of flow in circular clarifierswith and without swirl, 1994, J.of Hydr.Engrg ASCE, 120, 4-21.

33. Stamou, A. L Prediction of hydrodynamic characteristics of oxidationditches using the k-e turbulence model, Proceedings of 2nd Intern.Symposium on Engineering Turbulence Modelling and Measurements,Florence, Italy,pp. 261 to 271, 1993.

34. Stamou, A. I. Prediction of the pollutant concentration field in oxidationditches, Proceedings of the International Conference on Water Pollution'93: Modelling, Measuring and Prediction, Milan, Italy, pp.493 to 500, 1993.


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