Transaction B: Mechanical EngineeringVol. 17, No. 3, pp. 167{178c Sharif University of Technology, June 2010

Prediction of Hydraulic E�ciency ofPrimary Rectangular Settling Tanks Using

the Non-linear k � " Turbulence Model

B. Firoozabadi1;� and M.A. Ashjari1

Abstract. Circulation is created in some parts of settling tanks. It can increase the mixing level,decrease the e�ective settling, and create a short circuiting from the inlet to the outlet. All above-mentionedphenomena act in such a way to decrease the tank's hydraulic e�ciency, which quantitatively shows how ow within the tank is uniform and quiet. So, the main objective of the tank design process is to avoidforming the circulation zone, which is known as the dead zone. Prediction of the ow �eld and size ofthe recirculation zone is the �rst step in the design of settling tanks. In the present paper, the non-lineark� " turbulence model is used for predicting the length of the reattachment point in the separated ow ofa Karlsruhe tank. Then, the recirculation bubble size, which is out of the capability of standard turbulencemodels, is determined. Also, the e�ect of the separation zone size on the tank's hydraulic e�ciency isinvestigated.

Keywords: Settling tanks; Non-linear k � "; FTC calculation; Hydraulic e�ciency; Circulation.

INTRODUCTION

Settling tanks have been used for separating oatingparticles from the main ow. To optimize the operationof these tanks, it is required that a quiet ow of uid be formed in the tank in such a way thatthe sedimentation process be performed in the bestway. However, creation of the dead zone and theshort circuiting of the inlet to the outlet disturb thequietness of ow and leave negative in uences on theperformance of the tank. To prevent these phenomena,some ba�es are used in several parts of the tank.By optimizing the application of the ba�es in thetanks, the size of the recirculation zones is reducedand the plug ow percentage, which is the ideal owfor clari�ers, is increased. In fact, several parameters,e.g. location and type of in ow and e�uent, locationand size of ba�e, and rate of sludge withdraw couldin uence the e�ciency of settling tanks. Then, thebest way to determine their order of magnitude is todevelop a suitable mathematical model. Mathematicalmodels have been mainly developed, since incorpora-

1. School of Mechanical Engineering, Sharif University of Tech-nology, Tehran, P.O. Box 11155-9567, Iran.

*. Corresponding author. E-mail: �roozabadi@sharif.edu

Received 5 May 2009; received in revised form 6 December 2009;accepted 20 April 2010

tion of the standard k � " turbulence model in thesettling tank simulation by Rodi [1]. By using thismodel, researchers could simulate ow �elds withoutany requirement of especially empirical expressions.But, Stamou et al. [2] and many other researchershave shown that obtained results through applyingthe standard model are only qualitatively comparablewith experimental data. Only in the work of Celik etal. [3], for calculating Flow-Through Curves (FTC) ofa Windsor tank, did the predicted results �t excellentlywith experimental measurements. However, the reasonfor this exceptional compatibility was recognized aftermore consideration by Adams and Rodi [4]. In fact,introducing numerical di�usion, because of using ahybrid scheme incorporated by Celik et al. [3], causedthe compatibility, so their obtained results were unac-ceptable. After �nding this fact, the team conductedtheir research on the Karlsruhe basin and tried toconsider the ability of the k � " model by creatingmore complicated ow patterns. They concluded thatthe standard k � " model is not capable of preciselypredicting the size of the recirculation zone and velocity�eld in the region, which strongly a�ects the tank'se�ciency. To overcome the above weakness of thestandard model, Adams and Rodi [4] carried out theircalculations with a curvature modi�cation to the k� "model of Leschziner and Rodi [5]. This model had

168 B. Firoozabadi and M.A. Ashjari

shown good results in calculating the separation pointlength of backward-facing step ow. Nevertheless,overestimating the length of the recirculation zone inthe tank was slightly surprising. In this ow, theformation of streamlines with strong curvature causedthe modi�ed model to work improperly.

Linear k � " turbulence models, which, in fact,can be thought of as tensorially invariant mixing lengththeories, usually work quite well in the descriptionof unseparated turbulent boundary layers. Althoughin thin shear ows, normal Reynolds stresses do notplay an important role, they have a main role indescribing ows that involve recirculation zones. Inthese cases, incorporating the linear k � " models,which are completely incapable of accurately predictingnormal Reynolds stresses, is not a very appropriatemodel.

Preliminary research taking into account the ef-fects of non-linear terms and modifying the k�" modelwas undertaken by Launder and Ying [6] and Rodi [7].However, the obtained models were developed in sucha way that general invariance was not exhibited and,hence, good results only showed for special ows. Itis very interesting that the main steps in developinga general non-linear turbulence model had been takenmany years ago by Rivlin [8] and Lumley [9]. Theyhad focused on a clue whereby there exist strikingsimilarities between the main turbulent ow of a New-tonian uid and the laminar ow of viscoelastic uids.Continuation of their method by further researcherscaused the desired model, known as non-linear oranisotropic k�", to be obtained by Speziale [10]. A fewyears later, Launder [11] o�ered a very complete formof the model that contained higher order terms. Inaddition, these non-linear models have all advantagesof the linear k � " model; they can better describe thenormal stresses and can be incorporated into most k�"model computer codes in a relatively simple manner.

In this paper, some numerical problems that occurthrough the appearance of non-linear terms in theanisotropic turbulence model will be introduced. Inaddition, the size of the recirculation bubble in theKarlsruhe basin will be determined by using the non-linear k � " model and comparing it with Adams andRodi's [4] experimental measurements. Finally, theperformance of the non-linear model in predicting thehydraulic e�ciency and FTC of the tank, which arestrongly a�ected by the velocity �eld and recirculationsize, will be investigated.

MATHEMATICAL MODEL

The steady state, two-dimensional ow in the Karl-sruhe basin is determined using time averaged conser-vation equations. The Navier-Stokes and continuityequations for Newtonian incompressible uid are ex-

pressed as:

@u@x

+@v@y

= 0; (1)

u@u@x

+ v@u@y

=� @~p@x

+ ��@2u@x2 +

@2u@y2

�� @�11

@x� @�12

@y; (2)

u@v@x

+ v@v@y

=� @~p@x

+ ��@2v@x2 +

@2v@y2

�� @�12

@x� @�22

@y: (3)

where u and � are the mean velocity componentsin the x and y directions; ~p is the modi�ed meanpressure de�ned by Speziale and Thangam [12]; �11,�22 and �12 are normal and shear components of theReynolds stresses, respectively, and � is the kinematicviscosity of the uid (here, it is water at 20�C).The Reynolds stresses could be modeled via an eddyviscosity assumption based on Speziale [10] in the form:

�ij =23k�ij � 2C�

k2

"Sij � 4CDC2

�k3

"2

��Soij � 1

3Sokk�ij + SikSkj � 1

3SklSkl�ij

�; (4)

where subscripts i; j = 1 and 2 stand for x and y,respectively; k is the kinetic energy of turbulence; "is the turbulent dissipation rate, and C� is a dimen-sionless constant that is taken to be 0.09, as o�eredby Rodi [1]. CD is the dimensionless constant, de�nedas 1.63, based on calibration of the non-linear modelwith Laufer's [13] experimental data for fully developedturbulent channel ow; �ij is the Kronecker delta, andSij , the mean rate of strain tensor, is expressed by:

Sij =12

�@ui@xj

+@uj@xi

�; (5)

Soij is the frame-indi�erent Oldroyd derivative deter-mined by :

Soij = uk@Sij@xk

� @ui@xk

Skj � @uj@xk

Ski: (6)

It is clear that at the limit of CD ! 0, the standardk � " model is recovered from the non-linear model.This is the same as the e�ective viscosity hypothesis ofBoussinesq, which relates the Reynolds stresses solelyto the rates of strain of the uid. This formula hasbeen used with considerable success by Ng [14] andRodi [15] for free shear ows, in conjunction with

Hydraulic E�ciency of Primary Settling Tanks 169

turbulence models, for k and ". It has been observed,however, that the Boussinesq hypothesis fails in anumber of applications, such as separated turbulent ows. Bradshow [16] has stated that this failure is dueto the form of the stress strain relation rather thanthe inapplicability of the eddy viscosity approach. Inparticular, the third term in Equation 4 corrects thefundamental weaknesses of the Boussinesq relationship.As shown by Pope [17], the new relationship hasa strong ability to capture normal stress anisotropy,high sensitivity to secondary strains, and an abilityto generate excessive turbulence at separation zones.Detailed tests by Speziale [10], Suga [18], and Craftet al. [19] have shown that the e�ect of the last termin Equation 4 is a signi�cant improvement in theprediction of the reattachment length of the separationregion behind the backward facing step, with resultssimilar in accuracy to those obtained using a Reynoldsstress model. However, the main advantage of thenon-linear model over more sophisticated turbulencemodels is that the time consuming solution of thestress equations, in the form of partial di�erential oralgebraic, is not needed. In addition, the inter-relationbetween strain and stress has been retained withinthe di�erential equation, which increases the numericalstability of the model. Our numerical experiences haveshown that the non-linear model of Speziale [10] hasa strong capability of predicting separated ow withinsettling tanks, while its numerical implementation andstability is, approximately, as simple and as good as thestandard model. Nevertheless, some �ner grids and,consequently, more CPU time are needed to deal withnon-linear terms.

It is worth mentioning that k and ", in the non-linear model, are calculated using the same transportequations incorporated in the standard k�" model [1].

Through Equations 1 to 6, the ow �eld in thebasin could be solved and, hence, the tank's modelis fully de�ned hydrodynamically. To complete thetank's modeling, the determined velocity �eld mustbe introduced into the transport equation of the dyesuch that the concentration can be calculated. Bysupposing that the dye has no in uence on the ow�eld, and is dispersed only via di�usion and convectionmechanisms, the transport equation will be:

@C@t

+@uC@x

+@vC@y

=@@x

��t@C@x

�+@@y

��t@C@y

�; (7)

where C is the mean dye concentration and �t, theisotropic eddy di�usivity coe�cient, is proportional tothe eddy viscosity, �t = �t

�c , according to the Reynoldsanalogy theory for mass and momentum transport.Here, proportional coe�cient �c is the Schmidt numberand is 0.7, as proposed by Launder [11], for freeturbulent ows.

INITIAL AND BOUNDARY CONDITIONS

The hydrodynamics part of the problem is steady stateand the initial condition for the velocity �eld is notvery important. Nevertheless, for calculating the tank'sFTC, no dye must exist, initially, all over the tank (C =0).

By knowing the velocity at the �rst node abovethe solid boundary, the shear velocity is calculated viathe law of the wall, in the standard two-layer formdeveloped by Launder and Spalding [20], and the wallis taken as being smooth. Formally, the wall functionmust not be applied to separated-turbulent boundarylayers, and many researchers like Avva et al. [21] triedto modify it in several ways. However, since the ow�eld is solved iteratively, and the separated regionsoccupy a small space, con�ned only at the tank corner,major errors do not appear to the overall ow �eld.After calculating the shear velocity, the values of k and" at the adjacent node above the wall are determinedby assuming local equilibrium for turbulent productionand dissipation [1]. All boundary conditions at theinlet, outlet, and free surface of the model tank areapplied as recommended by Celik et al. [3].

NUMERICAL SOLUTION PROCEDURE

A new computer code, in FORTRAN 90, is developedto solve hydrodynamics and mass distribution equa-tions, numerically. The use of staggered grids anda control volume method is the main characteristicof the new code. However, the pressure �eld iscorrected at each step by the SIMPLEC algorithmintroduced by Patankar [22]. To enhance numericalstability, the convection terms are approximated byusing a hybrid (upwind/central) scheme. In addition,an under-relaxation factor for all dependent variablesmust be taken to control the strong tendency fordivergence. Nevertheless, as the turbulent productionterm in transport equations for k and " [1] is non-linear and the velocity gradients at primary stepsare so digressive, the eddy viscosity must be limitedto less than 2500 � and under-relaxed in each itera-tion.

To ensure that the non-linear turbulence modelis applied correctly in the new code, a fully developedturbulent ow in a channel, as a simple test case, issimulated. The normal stresses obtained by the codeusing di�erent models are compared with experimentaldata of Laufer [13] in Figure 1. In contrast withthe non-linear model, the linear model has the sameestimation of both normal stresses. These resultssimultaneously validate both the non-linear model andthe developed code itself.

In order to discretize Equation 4, it is requiredthat the Reynolds stresses be determined at the main

170 B. Firoozabadi and M.A. Ashjari

Figure 1. Normal Reynolds stresses in fully developedturbulent channel ow.

nodes where ~p; "; k; and C are stored. But, at the mainnodes near the boundary, the Oldroyd derivative termscould not be approximated via a central di�erencescheme, which has a second-order error. Hence, theReynolds stresses are extrapolated there. The validityof this simpli�cation could be explained by noting thegeneral behavior of turbulent ows near the boundaryand linear variations of the stresses, there, in fullydeveloped turbulent channel ows (Figure 1). In addi-tion, for the staggered grid [23] used in the calculations,the shear stresses must be known at the cell facesinstead of the main nodes where scalar variables arestored. Interpolating could easily solve this problem,but the stress consistency between the two adjacentcells must not be forgotten. This means that foursurrounded main nodes are required to interpolate theshear stress at the cell face.

To control the strong divergence, because of theapplication of the non-linear k�" model, two strategiesare taken into account in the solution. First, thecalculated ow �eld, by the linear k � " model, isintroduced as an initial condition. Second, the e�ects ofnon-linear terms in the anisotropic model are graduallycontributed to the solution process by imposing alimitation condition on them. The upper bound ofthis limitation is de�ned by comparing the non-linearterms with their corresponding linear terms (i.e. tworight hand side terms) in Equation 4. It is worthnoting that the modi�ed terms in Equation 4 havemore dispersive features than the favorable dissipativeone. Thus, in solving the ow �eld, by using the non-linear k� ", and with the initial condition obtained bythe standard k � ", the averaged-mean-square errorsresulting from mass imbalances in control volume willnot be drastically reduced. Hence, the calculationprocess is continued until the di�erence in length of

the separation point of the successive iterations is lessthan 1%.

In order to calculate the FTC of the tank, Equa-tion 7 is discretized using a fully implicit time dis-cretization method proposed by Roache [24]. Althoughthis method is only �rst order, an accurate calculationcould be achieved by choosing a reasonable time step,which should be determined case by case. Nevertheless,dominant errors of approximating convection termswhich have shown themselves as \so called numericalor false di�usion" are still present, which virtuallyincrease the physical di�usion. In fact, as Roache [24]has shown, the solved di�erence equations will give thesolution of the equation below instead of Equation 7,when convection terms are approximated by using thehybrid scheme:

@C@t

+@uC@x

+@vC@y

=@@x

�(�t + �num)

@C@x

�+

@@y

��t@C@y

�+HOT; (8)

where HOT stands for Higher-Order Terms. �num isthe numerical di�usivity from the numerical approxi-mation errors. Numerical di�usion is neglected in the ydirection in Equation 8 because the ow in the tank isnearly unidirectional and the cross-stream convectionis small.

A practical way to eliminate any numerical di�u-sion, especially in unsteady problems, is to incorporatemore accurate di�erencing schemes. However, usinghigher-order schemes has caused stability problemsoften called wiggles in literature, which limit theapplication of more accurate schemes. So, as a �rstapproximation, the most popular second-order QUICKis recognized as being suitable for the FTC calculationof the Karlsruhe tank. In addition, due to an increasein the bandwidth of the linear equations system byusing the standard QUICK scheme of Leonard [25] theQUICK scheme of Hayase et al. [26] is used here.

FLOW FIELD PREDICTION USING THEMODEL

Figure 2 shows the dimensions of the Karlsruhe modeltank, which is selected for ow consideration at Re =10000 (de�ned with respect to hi, inlet slot high). Itsexperimental measurements were given by Adams andRodi [4].

For ow computations of the surface dischargecase, Si=h = 0:91, a 75 � 35 grid was su�cient togive grid independent solutions. This could be furtherexamined through Figure 3, which shows root meansquare errors in di�erent entrance cross sections for u-velocity and eddy viscosity pro�les. Errors for any

Hydraulic E�ciency of Primary Settling Tanks 171

Figure 2. Dimensions of Karlsruhe basin; in surfacedischarge case Si = 10 cm and in submerged dischargecase Si = 6:468 cm.

Figure 3. u-velocity and eddy viscosity error in di�erentcross sections for surface discharge case. (a) Linear; (b)

Non-linear model (E(X) =

s1N

NP1

(X �Xref)2).

examined grid are calculated relative to a selectedreference grid, for instance 150 � 70. A solution isconsidered grid independent if its error drops belowa prescribed value, chosen here as 0.02. Accordingto this criterion, a 75 � 35 grid is su�ciently �nefor both linear (Figure 3a) and non-linear (Figure 3b)models. Figure 4a shows the streamlines by using a

Figure 4. Calculated ow �eld with 75� 35 grid forsurface discharge case. (a) Linear k � "; (b) Non-lineark � ".

linear k � " model. The reattachment length is thesame as that reported by Adams and Rodi [4]. Thispredicted length is only at the lower range of theexperiments (xr=h = 5:5 � 6:5). This can be relatedto the incapability of the linear model. In contrast, inthe ow �eld, which is calculated by non-linear k � ",as illustrated in Figure 4b, the length of the separationpoint is increased and reaches xr=h = 6:14. This isnow in the experimental range completely and showsclearly the e�ectiveness of the applied model.

However, the performance of the non-linear modelcould be seen more obviously if the submerge dischargecase, Si=h = 0:588, is considered. Due to thecomplication of this ow, a grid independent solutionis not achieved for both linear and non-linear modelswith the same grid size (Figure 5). This can alsobe seen in Figures 6 and 7 that show the calculatedstreamlines with two di�erent grids. Although usinga �ner grid with the linear k � " has not causedmuch di�erence in the ow �eld (Figures 6a and 7a),the di�erence in the size of the recirculation bubbleis considerable in the case of the non-linear model(Figures 6a and 6b). Therefore, it is inferred that whenthe non-linear k � " is applied for a complicated ow�eld, the corresponding size of the grid must be aboutfour times greater than the size of a grid that was givena grid independent solution with the linear k�" model.This will cause the computation time for the non-linear

172 B. Firoozabadi and M.A. Ashjari

Figure 5. Error of u-velocity and eddy viscosity indi�erent entrance cross sections for submerged dischargecase. (a) Linear and (b) Non-linear k � " model

(E(X) =

s1N

NP1

(X �Xref)2).

model to be dramatically increased. Table 1 showsthat when the same grid is used for both models (insurface jet), the non-linear model has a computationaltime approximately two times greater than the linearmodel. However, for a submerged jet, a �ner grid isrequired for the non-linear model, so that it has about30 times greater computational time. This could onlybe justi�ed in cases where high accuracy is needed e.g.in tank performance calculations (next section).

For the case Si=h = 0:588, two small and largeseparation zones exist in the ow. Their calculatedlength, using two models, is compared in Figure 7.Although the modi�cation in the size of the uppersmall separation zone is not so noticeable, the sizeof the lower large separation zone is increased fromxr=h = 4:37, for linear k � ", to xr=h = 6:0, fornon-linear k� ", and well approaches the experimentalvalue (xr=h)e = 6:17. Hence, the non-linear model,without any regard to the ow type and its com-

Figure 6. Computed streamlines with 75� 52 grid forsubmerged discharge case. (a) Linear model; (b)Non-linear model.

Figure 7. Computed streamlines with 150� 100 grid forsubmerged discharge case. (a) Standard k � "; (b)Non-linear k � ".

Hydraulic E�ciency of Primary Settling Tanks 173

Table 1. Computational time and separation length errorof di�erent grids.

Surface Discharge Case

Grid Size Linear k � "xr=h

ReattachmentLength Error

CPU Times(s)

19 � 9 5.10 9.0% 8

38 � 18 5.39 3.7% 38

75 � 35 5.49 2.0% 646150 � 70

(Reference)5.6 0.0% 9267

Non-Linear k � "19 � 9 5.65 8.9% 17

38 � 18 6.04 2.6% 77

75 � 35 6.14 1.0% 892150 � 70

(Reference)6.20 0.0% 18780

Exp.Adams et al.

[4]6.4 | |

Submerged Discharge Case

Grid Size Linear k � "xr=h

ReattachmentLength Error

CPU Time(s)

38 � 26 3.2 27.0% 74

75 � 52 4.32 1.3% 878

150 � 100 4.37 0.2% 13210300 � 200(Reference)

4.38 0.0% 88122

Non-Linear k � "38 � 26 3.83 36.7% 135

75 � 52 5.18 14.4% 1700150 � 100 6.00 0.8% 25000300 � 200(Reference)

6.05 0.0% 190000

Exp.Adams et al.

[4]6.2 | |

plexity, acts quite well in predicting the size of largerecirculation zones, which are more interesting. Themodel only fails in simulating small recirculation zoneswhere the curvature of streamlines is extraordinarilystrong. These small separation bubbles are not moreimportant, as they have negligible in uence on the owcharacteristics.

The reason why the standard turbulence modelis unsuccessful in predicting separated ows will beclearer when momentum equations are written in termsof the mean ow stream function, , (where u = �@ @y

and v = @ @x ) as:

u@@x�r2

�+ v

@@y�r2

�= (� + �t)r4

� @2 (�22 � �11)@x@y

� @2�12

@x2 +@2�12

@y2 : (9)

This equation shows that in cases where high velocitygradients are present, the linear k � " model is unableto predict the normal Reynolds stress di�erence, �22 ��11. This term contributes directly to calculating thestreamlines and their curvatures. It can be shownthat the linear model predicts the sum of the normalReynolds stresses as (�11 + �22 = 2�33), for any 2-D ows, that surely will not satisfy all cases. Inother words, the linear model has some problems incalculating the normal stresses.

Figure 8a compares the measured and calculateddimensionless streamwise velocity, u=uin, at severalcritical sections in the recirculation zone for the sub-merged discharge case (the case in which the inlet ow is from the middle height of the tank). Thesuperiority of the non-linear model, especially in theregions near the separation point, x=h = 5:58, 5:88 and6:17, is clearly apparent. Furthermore, Figure 8b givesadditional information about the status of turbulence

Figure 8. Dimensionless velocity (a) and turbulencekinetic energy (b) pro�les at several critical sections inrecirculation zone of Karlsruhe tank in submergeddischarge situation.

174 B. Firoozabadi and M.A. Ashjari

in the tank. In this �gure, the calculated dimensionlessturbulence kinetic energy (divided by the square meaninlet velocity, u2

in) using both standard and non-lineark � " models, is compared with the experimental dataof Adams and Rodi [4]. As expected, the highestturbulence levels are found in shear layers borderingthe separation zones (i.e. x=h < 6:2). Beyondreattachment, the turbulence level drops quickly toabout a constant small value due to the absence of anysigni�cant velocity gradients. This simply explains whythe standard k � " model has very poor predictions ofturbulence kinetic energy levels within the separationzone, while having a good performance out of thisregion. As shown earlier, the section, x=h = 5:58,for a submerged discharge case, is entirely out ofthe separation region when the linear k � " modelis incorporated. In contrast, the separation zone hasextended beyond this station for the non-linear k � "model and for measurements. Therefore, nearly goodagreement is observed between the non-linear k � "model and experimental data for this critical section.However, separation has ended at this section and,consequently, the turbulence level has dropped for thelinear k � " model.

CALCULATING THE FTC

The hydrodynamic performance of a tank is obtainedby injecting uorescent dye with the same densityas the internal water at a de�nite time in the inletand measuring the outlet dye concentration severaltimes. A curve that shows the dye concentrationvariation in the outlet with time is called a Flow-Through Curve (FTC). It is more convenient to presentthe FTC with dimensionless variables. Hence, timeis non-dimensionalized by the theoretical detentiontime, Tth = hl

q , where q is the inlet ow rate, andconcentration by Co = hl

min, where min is the total mass

of the dye to enter the tank.To calculate the FTC of the Karlsruhe tank, the

concentration transport Equation 7 must be solved byintroducing the computed ow �eld in the previoussection. In addition, for simplicity, the inlet boundarycondition is taken as C = 1 during the injection timeand C = 0 for the next times.

Figure 9a compares the calculated FTC for asurface discharge situation, with both hybrid andQUICK schemes, using two di�erent ow �elds ob-tained by linear and non-linear models, with Adamsand Rodi's [4] measurements. In this case, the peakconcentration predicted with the hybrid scheme andnon-linear k � " model is almost the same as thatmeasured, while the peak concentration predicted bythe QUICK scheme and the non-linear k � " modelis about 35% higher. However, this value is more

(about 60%) when a linear k � " model is used. Thediscrepancy between the numerically accurate QUICKcalculation and the measurements may be due to threereasons. The �rst possibility is that the incorporatedturbulence model in the hydrodynamic part cannotproduce su�cient mixing, i.e. eddy di�usivity as it isproduced in the experiment. According to Figure 9b,this is the case, but it does not solely justify thepoor prediction of the QUICK scheme. The levelof the turbulence uctuations predicted by the lineark � " model is generally lower than the measurementsand the non-linear turbulence model. Therefore, theFTC calculation for the ow �eld obtained by thenon-linear k � " model can reduce peak concentrationto about 25%. This is high, but not as much asexpected. The second possibility for the di�erencein peak concentration may be due to the inconsis-tency of the QUICK scheme with the physics of theproblem. The QUICK scheme is often used in steadystate calculations (also it has been used for transientcomputations by some researchers), while many otherschemes have been proposed originally for transientcalculations. To discover the e�ect of unsteadiness inthe calculations, we have repeated the FTC calculationwith the third order accurate QUICKEST schemedeveloped originally by Leonard [25]. To avoid the oc-currence of non-physical numerical oscillations, a mod-i�ed ULTIMATE algorithm has also been used [27].The result (Figure 9b) is observed as a lowering inpeak concentration by an amount of 8% relative tothe QUICK scheme. Hence, another possibility mayalso exist. Based on the statements of Adams andRodi [4], three-dimensional motions were observed inthe experimental tank, in spite of their attempt to keepthe motion two-dimensional through the large aspectratio of the tank. De�nitely, additional mixing due tothese three-dimensional motions is not reproducible byusing the two-dimensional model.

Figure 9c, for a submerged discharge case, showsthat the peak concentration is highest for the lineark � " with the QUICK scheme and lowest for thenon-linear model with the hybrid scheme. In thissituation, when the size of the recirculation zone isenlarged, the peak concentration decreases, as shownpreviously. But, the passing time of peak concentrationdecreases.

Figure 9d shows a comparison between the FTCof the two di�erent cases, which were calculated usingprecise ow �elds and a QUICK scheme without nu-merical di�usion. In the submerged discharge case, theFTC is changed in such a way that it approaches theideal step FTC that could be considered as a sign ofhigher hydraulic e�ciency.

Through the FTC, some important parameters,such as the level of short-circuiting, the mixing levelin uenced by both di�usion and/or recirculation zones,

Hydraulic E�ciency of Primary Settling Tanks 175

Figure 9. FTC for (a, b) surface and (c) submerged, discharge cases and (d) comparison with each other.

and overall e�ciency, could be calculated. The time ittakes for the �rst dye to appear at the outlet is de�nedas the short-circuiting level and is shown by t0 and t10indices (tn is the time which takes until n percent ofthe total injected dye into the tank is passed throughthe outlet). The value of these indices depends on boththe ow �eld and the mixing level. When the values arelow, one may conclude that short-circuiting exists andit shows a false designing of the tank. The mixing levelin the tank, de�ned by t75 � t25, t90 � t10 and t90=t10,is increased by increasing the di�usion coe�cient andthe size of the recirculation bubble. Nevertheless, theperformance of the tank is increased by the former anddecreased by the latter. t50 and tmax, the passing timeof peak concentration, are indices to express the overalle�ciency of the tank.

All the computed FTC characteristic indices ofthe Karlsruhe tank are summarized in Table 2. The

short-circuiting indices are decreased or, in otherwords, the level of short-circuiting is increased byincreasing the separation bubble (compare columns 1with 2 or 4 with 5) and the virtual di�usion coe�cientcreated by additional numerical di�usion (comparecolumns 2 with 3, or 5 with 6). In addition, Table 2shows that the tank's mixing level is increased when theabove-mentioned factors increase. This high mixingdisturbs the still circumstances, which are requiredfor e�cient settling. Altogether, it is inferred fromTable 2 that increasing the recirculation size causesa high level of short circuiting and mixing. Theresult of these complicated events can be seen onthe reduction of t50 and tmax indices that indicatetank performance. Conversely, the e�ciency of thetank is not more sensitive to the di�usion coe�cient;however, the mixing level in the tank is increased byits increasing (see columns 2, 3 or 5, 6 of Table 2).

176 B. Firoozabadi and M.A. Ashjari

Table 2. FTC indices of Karlsruhe tank.

Si=h = 0:588 Si=h = 0:91

FTC IndexStandardk � "Quick

Non-Lineark � "Quick

Non-Lineark � "

HYBRID

Standardk � "Quick

Non-Lineark � "Quick

Non-Lineark � "

HYBRID

Exp.Adams

et al. [4]Short

Circuitingt0 0.67 0.653 0.59 0.62 0.584 0.407 0.45

t90 0.865 0.843 0.828 0.73 0.713 0.66 0.65

Mixing

t75 � t25 0.225 0.258 0.276 0.228 0.253 0.307 0.32

t90 � t10 0.513 0.577 0.601 0.554 0.604 0.679 0.59

t90=t10 1.593 1.684 1.726 1.755 1.847 2.029 1.91

E�ciency

tmax 0.965 0.942 0.953 0.811 0.797 0.797 0.78

t50 1.023 1.01 1.018 0.862 0.859 0.872 0.87

CONCLUSION

This paper has presented the main reasons for de-�ciency in the standard k � " turbulence model inprediction of the separation point length of the owin the Karlsruhe tank. The non-linear k� " model wasused to overcome the weakness of the linear model andto calculate the size of the recirculation zone exactly.Comparing the computed streamwise velocity pro�leswith measurement data in several critical sectionsshows that the non-linear model is more e�ective. Inaddition, incorporating the velocity �elds, obtainedby the two di�erent turbulence models, shows that aprecise ow �eld is the main requirement for calculatingan accurate FTC. If the size of the separation bubblesin the Karlsruhe is predicted slightly larger, a reductionwill appear in the tank's overall e�ciency. However,this reduction is not very notable because the size ofthe recirculation zone is small, with respect to thetank's volume, and strong 3-D e�ects were present inthe tank.

To consider the complicated e�ects of the recir-culation zones, the non-linear k � " model was usedto calculate the FTC. This is reasonable because thenon-linear model has no increased computation costsand can be incorporated simply into codes with thestandard k � " model.

ACKNOWLEDGMENT

The authors would like to express their gratitude andsincere appreciation to Dr. M.T. Manzari, associateprofessor of Sharif University of Technology for hisvaluable comments to overcome the numerical prob-lems of this project.

NOMENCLATURE

C mean dye concentrationCo reference concentrationCD coe�cient of non-linear terms in

Reynolds stresses formulaC"1; C"2 experimental constants in " transport

equationC� dimensionless constant in Reynolds

stresses formulaE(X) root mean squares error of variable X

in a selected cross sectionh depth of basinhi width of inlet slotk kinetic energy of turbulenceL length of basinmin total injected mass of dyen normal distance from the solid wallN number of data in each cross section to

calculate E(X) (� 20)P turbulence production term~p modi�ed mean pressureq inlet ow rateRe Reynolds number de�ned with respect

to inlet slot highSi location of inlet slot above tank bottomSij mean rate of strain tensorSoij frame-indi�erent Oldroyd derivativeTth theoretical detention timet timetmax passing time of the peak concentration

Hydraulic E�ciency of Primary Settling Tanks 177

u; v mean velocity components in the x andy directions

uin inlet velocityx; y streamwise and vertical directionxr separation point lengthyp normal distance to the wall from wall

adjacent grid point�ij Kronecker delta�t isotropic eddy di�usivity coe�cient�num numerical di�usivity" turbulent dissipation rate� kinematic viscosity�t isotropic eddy viscosity�c turbulent Schmidt number for dye�"; �k Schmidt number for k and "�xx; �yy normal components of Reynolds

stresses�xy shear component of Reynolds stress mean ow stream function normalized

by ow rate (q)

Subscripts

1; 2 x and y directions, respectivelyn percent of total injected dye which has

passed through outlete experimental dataref selected reference grid

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BIOGRAPHIES

Bahar Firoozabadi is associate professor in theschool of mechanical engineering at Sharif Universityof Technology, Tehran. Her research interests are

uid mechanics in density currents, presently focusingon bio uid mechanics, and porous media. Shereceived her PhD in mechanical engineering also atSharif University. She teaches uid mechanics andgas dynamics for undergraduates, and viscous ow,advanced uid mechanics, continuum mechanics andbio uid mechanics for graduate students.

Mohammad Ali Ashjari was born in 1977 at Tabriz,centre of East Azarbayjan province, Iran. His academicprogram has been started since 1996 when he hasentered Tabriz University to study Mechanics of Fluids.He had continued his education in Sharif Universityof Technology (SUT) in M.Sc. degree. He hasreceived PhD degree in Thermo-Fluid from SUT in2008. Dr. Ashjari's interest research �eld is developingand introducing novel numerical methods for uid ow simulation under complicated conditions includingturbulence and multi-phase porous media. He is nowfaculty member of Jolfa's Azad University.