Approximate Method for Designing a Primary Settling Tank for WastewaterTreatment

Gloria Martınez-Gonzalez, Herbert Lorıa-Molina, David Taboada-Lopez,Francisco Ramırez-Rodrıguez, Jose Luıs Navarrete-Bolanos, and Hugo Jimenez-Islas*

Departamento de Ingenierıa Quımica-Bioquımica, Instituto Tecnologico de Celaya, AVe. Tecnologico y GarcıaCubas s/n, Celaya, Gto. 38010, Mexico

An approximate method based on an analogous Fick’s law mass transport model was derived to predict theconcentration profiles of suspended particles in a primary sedimentation tank for wastewater treatment as afunction of time and column height. A pilot sedimentation column was constructed to test wastewatersedimentation and obtain the necessary data for the proposed model. The examined variables included thetotal suspended solids concentration, the sedimentation height, and the time elapsed. Computer code wasdeveloped to estimate the dispersion coefficient present in the sedimentation model from experimental datavia least-squares regression, resulting in a relative error of 3.705%. The proposed model was validated bygood agreement with reported data in the literature. The basic dimensions for designing a primary sedimentationtank were obtained based on the experimental data from wastewater samples obtained by a wastewater collectorlocated in the industrial zone of Celaya City, Mexico. This methodology can be successfully applied fordesigning primary sedimentation tanks for wastewater treatment facilities.

1. Introduction

Sedimentation is a crucial stage of domestic and industrialwastewater treatment in which suspended particles are separatedfrom a liquid by gravity. Consequently, sedimentation is oneof the most frequently used unit operations for the clarificationof residual water. The physical phenomenon associated withthe gravitational precipitation of solid particles in a liquid hasbeen widely studied. The equation describing the velocity ofsedimentation, formulated by Stokes in 1851,1 is the startingpoint for analyzing sedimentation. From this incipient formula-tion, diverse assays have been performed to find a mathematicalmodel to explain sedimentation. Hazen2 introduced the surfaceloading concept in 1904 and analyzed sedimentation factors forsolid particles contained in diluted water solutions. In 1952,Kynch3 proposed a kinetic theory of sedimentation based onconcentration changes in a suspension. In recent years, studieshave been conducted examining the properties of sedimentationprocesses for ideal and flocculating suspensions at Universidadde Concepcion in Chile and Universitat Stuttgart in Germany.4

In spite of these advances that are proposed to unify studies onsedimentation of dispersed and flocculating suspensions, thereare not suitable equations to simulate the settling phenomenonin wastewater archetypes and facilitate the design of sedimenta-tion tanks.4 Therefore, exhaustive experimental measurementsare required for the construction of solid-removal curves forthe calculation of dimensions for primary settlings.5 Addition-ally, several experiments are required to obtain solid-removalcontour plots at different heights and times, which are alsorequired for construct charts that describe the total solid-removalpercentage in the tank at a given time. This procedure islaborious because it is necessary to have a large amount ofexperimental data to obtain plots with acceptable accuracy.6

Rigorous models based in computational fluid dynamics(CFD) are used to predict flow patterns and suspended soliddistributions within sedimentation tanks.7,8 These are normallyused to find the relationship between the tank hydraulics and

the process efficiency. The use of CFD-based models has notbeen common due to the inherent complexity of the correctedNavier-Stokes equations for turbulent flow8,9 and the costsassociated with the specialized hardware and software required;however, there have been reports of simplified models includingempirical parameters.10-12 Currently, few industrial organiza-tions use CFD techniques to study flow phenomena in theirwastewater treatment facilities due to the high cost of com-mercial licenses of CFD software.13

Thus, the objective of this study was to develop an ap-proximate and feasible mathematical model that allows for theprediction of concentration profiles of suspended particles basedon time and height measurements in a sedimentation column,without the need for conventional experimental-plottingprocedures.

Model Description. Wastewater is considered as an incom-pressible and Newtonian fluid within the sedimentation columnshown in Figure 1, containing particles of equal size and density(component A). These particles flow toward the bottom of thecolumn by gravity. The settling velocity is assumed to beconstant, and flocculation effects are assumed to be negligible.Equation 1 describes a microscopic mass balance using Fick’slaw adapted to macrodispersion, as reported by Johnson andDePaolo,14 Beg et al.,15 and Ginn et al.16 The dispersioncoefficient, DE, includes turbulence effects caused by particlesand gravity and was considered constant. Thus, the continuityequation for component (A) was as follows.

where v ) Vzk.Additional assumptions for the proposed sedimentation model

are as follows:1. Solid dispersion is equal for particles of equal size

throughout the liquid.2. Solid dispersion occurs in the vertical direction, z; thus,

the particle concentration is only a function of the time andcolumn height, CA ) CA(z, t).

* To whom correspondence should be addressed. Phone: +52 (461)6117575. Fax: +52 (461) 6117979. E-mail: hugo@itc.mx.

∂CA

∂t+ (v · ∇CA) ) DE∇2CA + RA (1)

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10.1021/ie801869b CCC: $40.75 2009 American Chemical SocietyPublished on Web 07/24/2009

3. There are no chemical reactions, RA ) 0; at t ) 0, CA )CA0; at z ) L and t > 0, CA ) 0. At z ) 0, the solid flux isequal to zero because the bottom of the column is closed.

4. Radial dispersion is negligible due to a particle densitygreater than 1 and a height/diameter ratio greater than 10.

On the basis of these assumptions, eq 1 reduces to eq 2.

The term Vz is referred to as the terminal velocity of particlesand takes into account Stokes’s law; for its calculation, it wasnecessary to define values for FS, FL, d, and µL. FS was definedas 2.65 kg/m3, a typical density of sand;5 FL and µL at 20 °Cwere defined as 1000 kg/m3 and 0.001014 kg/m · s, respectively;and, the particle diameter, d, was defined as 0.0001 m (100µm), an average value based on reported sedimentationprocesses.6,19

Here, Vz was obtained from Stokes’s law:

In Equation 2, the velocity, Vz, is negative due to effects fromthe model solution, which is also considered by Bustos et al.1

in their proposed dynamic models. Equation 2 has been obtainedby Pritchard17 and Mucha et al.18 using their own referencesystem, which is used in this work.

Initial conditions:

Boundary condition 1:

Boundary condition 2:

The Reynolds number was obtained from eq 7. Since Re <2, the use of Stokes’s law was validated.

2. Experimental Section

A collector was located at the industrial zone of Celaya City,Mexico, where six wastewater samples of 30 L were taken everytwo days for a period of twelve days.5,6 After sampling, we ranan assay in a pilot column with a height of 1.8 m and a diameterof 0.1524 m, as suggested in the literature.19 The column wasconstructed with five sampling ports equally distributed alongthe column. In order to avoid water spillage, a funnel was placedat the top of the column. At the bottom of the column, therewas a 1-in. cavity for sludge removal after each experiment.The column and accessories were constructed with sanitaryPVC. In each assay, samples were removed from each port attime intervals of 5, 30, 45, 60, and 90 min.

The developed model was used to design a sedimentationtank for the wastewater collector mentioned above. This wasintended to suggest an alternative solution to the contaminationproblems existing in this region.20

The experimental procedure was carried out as follows:1. For each sample, the initial suspended solid concentration

(SST0 at t ) 0) was determined according to the MexicanOfficial Norm NMX-AA-034-SCFI-2001.

2. The column was filled with residual water, and beforetesting, the particles were maintained in a uniform suspensionby injecting compressed air.

3. The column was operated for 90 min before the air inletwas closed. Samples were taken from each sampling port at 5,30, 45, 60, and 90 min.

4. The suspended solid content was measured for each sample.The following dimensionless groups were proposed to non-

dimensionalize eq 2:

Substituting these groups results in a dimensionless form of eq 2:

where the dimensionless coefficients are given by

and the boundary conditions become

Using Kynch’s3 boundary condition at � ) 1, we have

and the following initial condition:

3. Results and Discussion

The suspended solid concentrations measured at differenttimes and heights were standardized by dividing them by the

Figure 1. Schematic of analyzed system, including initial and boundaryconditions in the sedimentation column.

∂CA

∂t) DE

∂2CA

∂z2+ Vz

∂CA

∂z(2)

Vz )g(FS - FL)d2

18µ) -0.008859 m/s (3)

t ) 0, CA(z, 0) ) CA0, 0 < z < L (4)

z ) 0,∂CA

∂z |z)0 ) 0, t > 0 (5)

z ) L, CA(L, t) ) 0, t > 0 (6)

Re ) |dVzFL

µL| ) 0.873699 < 2 (7)

� ) zL

τ ) ttC

C* )CA

CA0(8)

∂C*∂τ

) R∂2C*

∂�2+ �∂C*

∂�(9)

R )DEtC

L2(10)

� )VztC

L(11)

� ) 0,∂C*∂� |�)0 ) 0, τ > 0 (12)

� ) 1, C*|�)1 ) 0, τ > 0 (13)

τ ) 0, C*(�, 0) ) 1, 0 < � < 1 (14)

Ind. Eng. Chem. Res., Vol. 48, No. 16, 2009 7843

initial concentration to obtain the fractions xR. Later, for everytime and height, the corresponding fraction was averagedaccording to the six initial concentrations as shown in Table 1.These average values were used to fit the dispersion coefficientin the sedimentation model given by eq 9.

The sedimentation model was solved via finite differencesmethod with 24 discretization nodes in the z direction and usingMarquardt’s algorithm21 to perform nonlinear estimation of DE.Mesh analysis revealed the suitability of this discretizationscheme. In order to fit the dispersion coefficient, Marquardt’salgorithm was linked to a code developed in FORTRAN 90,named OPTI-PF1, which solves the proposed model andcalculates the dispersion coefficient by comparing the modelpredictions (Table 2) with the experimental results shown inTable 1. The estimated value for the dispersion coefficient was0.002 723 63 m2/s with an average relative error of 3.707%.This value includes turbulence effects caused by the particlesand gravity. Subsequently, the fitted parameter was used in thesedimentation model to generate particle removal profiles basedon time and the height of the tank. Equation 9 was subsequentlyspatially discretized by central finite differences of second order,producing a system of ordinary differential equations. This setof differential equations was solved with a FORTRAN 90 codenamed PARFIN that solves a set of parabolic nonlinear partialdifferential equations using an explicit Runge-Kutta-Fehlberg method with an adaptive integration step.

Equation 9 was solved by selecting a characteristic time of5400 s, corresponding to the time in which all experiments wereperformed. The equation was then integrated from 0.0 to 1.0over 1000 intervals with a truncation error of 10-5.

The concentration variations of the removed solids are shownin Figure 2. With this information, an approximate design fora sedimentation tank can be proposed using both the modelresults and the conventional tank design procedure suggestedby Ramalho.5 Thus, the collected data from the sedimentationmodel has to be sorted according to depth, h, and the percentageof total removal can be obtained from eq 15.

where XRT is the total fraction of removed solids at a certaintime and xR0 is the fraction of solids remaining at the deepestpart of the sedimentation tank, when h ) H. The second termin eq 15 cannot be calculated directly by analytical ornumerical methods because an algebraic expression for h/Hbased on xR is not available. However, because equal intervalsof h/H are used, the integral presented in eq 15 can becalculated by reducing it to the total area of the graph shownin Figure 2 to perform the desired integration, which is shownin eq 16.

The total area of the plot is calculated by multiplying themaximum value for both variables, xR0 per xR and 1 per h/H.Simpson’s 3/8 rule with n ) 24 was employed to calculate thedefinite integrals in eq 16.

On the basis of the information provided by Tchobanoglousand Crites,18 the XRT values calculated by eq 16 were fitted bynonlinear regression to a hyperbolic function of the total removalfraction, XRT, as indicated in eq 17:

The parameters a and b were fitted as 5890.126 and0.94873322, respectively. By replacing both parameters a andb in eq 17, eq 18 is obtained. Equation 17 exhibits goodprecision for values of the total solid removal, superior to0.30, which is sufficient for sedimentation tank design.5

Table 1. Average Concentrations of Total Suspended Solids atDifferent Times and Column Heightsa

total suspended solids (kg/m3) at the indicatedcolumn height (given by the sampling ports).

time (s) 0.3 m 0.6 m 0.9 m 1.2 m 1.5 m 1.8 m

0 2.06 2.06 2.06 2.06 2.06 2.06300 2.02 2.00 1.94 1.86 1.75 0.001800 1.85 1.78 1.65 1.49 1.28 0.002700 1.74 1.66 1.52 1.36 1.14 0.003600 1.63 1.54 1.40 1.22 1.02 0.005400 1.47 1.37 1.25 1.08 0.90 0.00

a Average initial concentration of the total suspended solids ) 2.06kg/m3, standard deviation ) 1.14. Flow average ) 14.30 L/s, standarddeviation) 6.41.

Table 2. Predicted Concentrations Given by the ProposedMathematical Modela

total suspended solids (kg/m3) at the indicated column height.

time (s) 0.3 0.6 0.9 1.2 1.5 1.8

0 2.0566 2.0566 2.0566 2.0566 2.0566 2.0566300 2.0221 2.0016 1.9381 1.7604 1.2805 0.00001800 1.7981 1.7798 1.7231 1.5649 1.1381 0.00002700 1.6759 1.6587 1.6059 1.4585 1.0607 0.00003600 1.5619 1.5460 1.4968 1.3593 0.9886 0.00005400 1.3569 1.3431 1.3003 1.1809 0.8588 0.0000

a Average initial concentration of the total suspended solids ) 2.0566kg/m3. Flow average ) 14.3039 L/s.

Figure 2. Variation of the dimensionless removed solid concentrationCA/CA0 with the height of the settling column and the time ofsedimentation.

XRT ) (1 - xR0) + ∫0

XR0 hH

dxR (15)

XRT ) (1 - xR0) + ∫0

xR0 ( hH) dxR

) (1 - xR0) + AT - ∫0

1xR d( h

H)(16)

XRT ) ta + bt

(17)

XRT ) t5890.126 + 0.94873322t

(18)

7844 Ind. Eng. Chem. Res., Vol. 48, No. 16, 2009

For the subsequent design of sedimentation tanks, the surfaceload can be calculated from eq 19.

The surface load is fitted with respect to the fraction of totalsolid removal using eq 20 via least-squares regression and theMarquardt21 algorithm for nonlinear estimation of parametersc and m.

Equation 21 and Figure 3 show the results of the powerregression.

The surface load refers to the velocity of sedimentationexpressed in units of volume per area and time units. Next, thefraction of total suspended solids removed was used to calculatethe dimensions of the primary sedimentation tank for thecollector in the industrial zone of Celaya. Equations 22 and 23were used to calculate the detention time and the surface load,respectively. The results, corrected with safety factors5 for thedetention time, are shown below.

With the fitted values, design parameters for the sedimentationtank were obtained, including the area, diameter, volume, andeffective depth.

The surface area of the tank was calculated by eq 24, whereQ is the wastewater flow, which was 1235.865 m3/day in thiscase.

For a cylindrical tank, the diameter, Ds, can be obtained usingeq 25.

Additionally, the tank volume, Vs, is calculated with eq 26.

Finally, the effective depth of the tank is calculated accordingto eq 27.

The variable he represents the depth of the tank; this is basedon the time and surface load values previously fitted.22 Thevalues commonly used for primary sedimentation tanks areshown in Table 3. Table 4 shows the design results for theproposed sedimentation tank to be used in the industrial zoneof Celaya City, for a removal fraction of XRT ) 0.45.

To validate the obtained results, the developed model wasapplied to experimental data reported by Ramırez6 and Ecken-felder22 with good agreement. These results are shown in Table 5.

4. Conclusions

A method based on an analogous Fick’s Law model wasproposed to estimate suspended particle concentrations as afunction of the time and axial position in a settling tank. This

qs )Ht

f (19)

qs ) cXRT-m (20)

Figure 3. Power regression (s) for the calculated surface load (0) fromeq 19 as a function of the fraction of total removed solids.

qs ) 8.1704XRT-1.4505 (21)

tr ) 1.75t (22)

qr )qs

1.5(23)

As )Qqr

(24)

Table 3. Standard Information for Designing Cylindrical PrimarySedimentation Tanks5,18

depth units typical rank

detention time s 5400-9000surface load m3/m2 ·day 14-56diameter m 3-30percentage of removed solids % 40-70

Table 4. Design Parameters for Sedimentation Tanks Based on theResidual Water from the Industrial Zone of Celaya City, Mexico

parameter value units

flow 1235.865 m3/dayXRT 0.45detention time 4625.188 ssurface load 26.017 m3/m2 ·dayfitted detention time 8094.079 sfitted surface load 17.345 m3/m2 ·daysedimentation area 47.502 m2

diameter 7.777 mtank volume 115.778 m3

effective depth 2.437 m

Table 5. Comparison between the Design Parameters Reported inthe Literature and Those Obtained in This Work Using theProposed Sedimentation Model

Ramırez6 Eckenfelder22

parameter reported calculated reported calculated units

flow 7570.820 7570.820 3785.410 3785.410 m3/dayXRT 0.667 0.667 0.727 0.727detention time 3000.000 2524.357 6171.430 5250.369 ssurface load 48.895 59.010 25.725 28.738 m3/m2 ·dayfitted detention time 5250.000 4417.624 10800.000 9188.145 sfitted surface load 32.597 39.340 14.700 19.159 m3/m2 ·daysedimentation area 232.257 192.446 258.000 197.579 m2

diameter 17.196 15.653 18.124 15.861 mtank volume 460.032 387.095 473.176 402.557 m3

effective depth 1.981 2.011 1.834 2.037 m

Ds ) �4As

π(25)

Vs ) Qtr (26)

he )Vs

As)

Qtr

As(27)

Ind. Eng. Chem. Res., Vol. 48, No. 16, 2009 7845

method replaces the conventional experimental plotting method,requiring only the estimation of the dispersion coefficient, DE,using the least-squares method. With the prediction of isore-moval plots, the dimensions of an appropriate primary sedi-mentation tank can be calculated. The sedimentation model canbe used to predict the profiles of residual water concentration,although the initial concentrations can vary. For studyingwastewaters of other origins, it is only required simplesedimentation tests be carried out to calculate the dispersioncoefficient. The procedure of simulation and design is equiva-lent. The developed methodology was validated using data fromthe literature showing a good agreement.

Acknowledgment

The authors would like to acknowledge SES-DGEST forfinancial support.

Nomenclature

As ) area of the sedimentation tank, m2

A ) dimensionless parameterB ) dimensionless parameterC ) dimensionless parameterCA ) total concentration of suspended solids, kg/m3

CA0 ) total concentration of suspended solid at initial time, kg/m3

C* ) dimensionless total concentration of suspended solidsD ) average diameter of spherical particles, mDE ) dispersion coefficient, m2/sDs ) diameter of the sedimentation tank, mF ) conversion factor for units of length per time to units of volume

by area and time, 86 400 (m3/m2 ·day)/(m/s)G ) gravity acceleration, m/s2

H ) overall depth of the sedimentation tank, mhe ) effective depth of the sedimentation tank, mH ) depth of the sedimentation tank, mL ) overall height of the sedimentation tank, mM ) dimensionless parameterQ ) wastewater flow, m3/dayqr ) surface load, m3/m2 ·dayqs ) surface load, m3/m2 ·dayRA ) chemical reaction termRe ) Reynolds numberT ) detention time, stC ) characteristic time, str ) fitted detention time, sVs ) volume of the sedimentation tank, m3

Vz ) velocity of sedimentation in Stokes’s law, m/sxR ) fraction of the total suspended solids remaining in suspensionXRT ) fraction of the total suspended solids removedZ ) Cartesian coordinate, m

Greek Letters

µL ) viscosity of the liquid, kg/m · sFL ) density of the liquid, kg/m3

FS ) particle density, kg/m3

� ) dimensionless height of the sedimentation columnτ ) dimensionless time of sedimentation

γ ) dimensionless coefficient� ) dimensionless coefficient

Literature Cited

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(2) Hazen, A. On sedimentation. J. Am. Soc. Chem. Eng. 1904, 53, 45–71.

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(10) Jin, Y. C.; Guo, Q. C.; Viraraghavan, T. Modeling of Class I SettlingTanks. J. EnViron. Eng.sASCE 2000, 126, 754–760.

(11) Guo, Q. C.; Jin, Y. C. Estimating Coefficients in One-DimensionalDepth-Averaged Sediment Transport Model. Can. J. CiVil Eng. 2001, 28,536–540.

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(13) Brouckaert, C. J.; Buckley, C. A. The Use of Computational FluidDynamics for Improving the Design and Operation of Water and WastewaterTreatment Plants. Water Sci. Technol. 1999, 40, 81–89.

(14) Johnson, T. M.; DePaolo, D. J. Reaction-Transport Models forRadio Carbon in Groundwater: The Effects of Longitudinal Dispersion andthe Use of Sr Isotope Ratios to Correct for Water-Rock Interaction. WaterResour. Res. 1996, 32, 2203.

(15) Beg, S. A.; Hassan, M. M.; Naqvi, M. S. M. Modeling of GasAdsorption Accompanied by Chemical Reaction in Down Flow Co-currentPacked Columns. Chem. Eng. Commun. 1998, 165, 41–66.

(16) Ginn, T. R.; Murphy, E. M.; Chilakapati, A.; Seeboonruang, U.Stochastic-Convective Transport with Nonlinear Reaction and Mixing:Application to Intermediate-Scale Experiments in Aerobic Biodegradationin Saturated Porous Media. Journal of Contaminant Hydrology. 2001, 48,121.

(17) Pritchard, D. The Rate of Deposition of Fine Sediment fromSuspension. J. Hydr. Engrg. 2006, 132, 533–536.

(18) Mucha, P. J.; Tee, S. Y.; Weitz, D. A.; Shraiman, B. I.; Brenner,M. P. A Model for Velocity Fluctuations in Sedimentation. J. Fluid Mech.2004, 501, 71–104.

(19) Tchobanoglous, G.; Crites, R. Tratamiento de Aguas Residualesen Pequenas Poblaciones; McGraw Hill: Santa Fe de Bogota, Colombia,2000; pp 114-126.

(20) Lorıa M. H. Obtencion de un Modelo Matematico de Sedimentacionpara Aguas Residuales. Tesis de maestrıa en Ciencias, Instituto Tecnologicode Celaya: Guanajuato, Mexico, 2005.

(21) Marquardt, D. W. An algorithm for least-squares estimation ofnonlinear parameters. J. Soc. Ind. Appl. Math. 1963, 2, 431–441.

(22) Eckenfelder, W. Industrial Water Pollution Control; McGraw-Hill:New York, 1989; pp 113-131.

ReceiVed for reView December 4, 2008ReVised manuscript receiVed June 24, 2009

Accepted July 20, 2009

IE801869B

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