INFLUENCE OF OFF-TAKE ANGLES ON FLOW DISTRIBUTION AT ...

INFLUENCE OF OFF-TAKE ANGLES ON FLOW DISTRIBUTIONPATTERN AT CONCAVE CHANNEL BIFURCATION

OBASI , N.L. , AGUNWAMBA , J.C. AND EGBUNIWE , N. 1 2 2

1. Department of Civil Engineering, Enugu State University of Science and Technology, Nigeria.2. Department of Civil Engineering, University of Nigeria, Nsukka, Nigeria.

ABSTRACTA physical model with meandering features was constructed and used to investigate theeffect of off-take angles on the flow distribution at a concave channel bifurcation. Sevendifferent off-take angles with varied main channel flow rates were used for the study.Predicting equations for the off-take discharge dependent on the off-take angles, mainchannel discharges, dispersion coefficients and Reynolds numbers were developed andcalibrated statistically. Results of the study and predicting equations showed that the off-take discharge increased positively with increases in off-take angles as well as mainchannel discharges. The developed empirical predicting equations for the off-takedischarge gave correlation coefficient values of 9.9974E-1 for both model equations withcorresponding standard errors of 9.754E-5 and 9.42E-5, respectively. It was observed thatthe predicted off-take discharge values from the model equations compared closely withthose of the study suggesting that off-take discharges for concave channel bifurcationscould be fairly predicted with the established model equations.

Keywords: Concave section, Off-take angles, Channel bifurcation, Flow rates.

NOTATIONS

1 2 3D , D , D Dispersion Coefficients for main and off-take channels.

i T Time after introduction of tracer

iC Tracer response concentration at selected points.θ Average flow time U Mean velocityL Distance traveled by tracer material before sample collection.s Normalized variance.

1 3Q , Q Main and off-take channel discharges

0 1 2, 3 4α , α , α α , α Regression constants; R - Regression correlation coefficientθ Off-take angle; Re Reynolds Number

m p1 p2Q , Q , Q Measured and predicted off-take discharges

INTRODUCTIONLeopold and Wolman [1] reported thatbifurcations are typical features in alluvialrivers as well as in estuaries and themorphological behavior of rivers atbifurcations is not yet a properlyunderstood phenomenon. A bifurcationoccurs when a river or stream splits intotwo branches and naturally it occurs when a

middle bar forms in a channel or adistributary carries flow from the mainriver.

In a meandering river/stream, theouter part of the curve is referred to as theconcave section of the channel curvature. Ifan intake is sited on the concave part of thecurve as is the case of the Obinna riverintake supplying water to Adani rice farm

46

47INFLUENCE OF OFF-TAKE ANGLES ON FLOW DISTRIBUTION PATTERN AT CONCAVE CHANNEL BIFURCATION

at Nsukka in Enugu State, Nigeria, it is saidto have a concave channel bifurcation. Theangle it makes with the outer part of themain channel is called the off-take angle.

According to Shettar and Murthy [2],at any channel bifurcation with a known

1upstream discharge (Q ), it is usuallydifficult to predict correctly the

2 3downstream discharges (Q and Q ) becausethey depend mostly on factors such as off-take angle, cross sectional areas, slope ofdownstream channels and shape of inletsection. They equally proposed that theflow which results when a channelbifurcates from the main channel has manycomplex features such as transverse motionaccompanying the main flow, extensiveseparation zone that develops in the branchchannel and re-circulation zone that appearsin the main channel at higher dischargeratios.

Research on division of flows hasreceived the attention of hydraulicengineers like [3-8] but most of their workswere restricted to finding out the dischargedistribution downstream of the bifurcationwhen the flow conditions upstream anddownstream of the bifurcation are known.Law and Reynolds [9] noted that both thetheoretical and experimental study of adividing flow in open channel reveals thatthe walls of the channels are mostly not theeffective boundaries of the flow regionresulting to the occurrence of separationzone. Taylor [10] conducted the firstcomprehensive study on a dividing flow inopen channel and suggested from theresults obtained that for a given angle ofintersection of the branch channel, it waspossible to correlate the discharge anddepth ratios with the ratios of the kineticenergy head to depth in the main channel.

Pattabhiramiah and Rajaratnam [11,12] studied the branch channel problem by

treating the branch flow as flow through aside weir of zero sill height while Law [13]used both analytical and experimentalinvestigations to study the features of thedividing flow.

Many fluid flow problems cannot besolved mathematically due to the complexnature of the equations and boundaryconditions inherent in them. It has beenhighlighted in most literature that even withmodern computing facilities; manycomplex fluid problems still defy completetheoretical analysis. In such situations,physical models seem to be the mostappropriate research tool to be employed.Essentially, physical models areconstructed to mirror the actual physicalbehaviour of the original phenomenon orprototype. The aim of this study is thereforeto ascertain the relative influence of the off-take angles on the flow distribution byusing the physical model to investigate theflow distribution pattern at a concavechannel bifurcation.

Model Construction and ExperimentalSetupThe model was constructed using the scaleratios of 1:50 for horizontal and 1:15 for thevertical. These scale ratios produced adistortion of 3.30. The model covered about200m of the river channel. The detailedplan views of the model and its crosssection are shown in Figure 1. Threestructural metal sheets having a width of30cm each were welded together to formthe base and side walls of the rectangularmain channel which represents fixed banksand bed of a natural river/stream. The mainchannel was 850cm long with straight andmeandering sections. Similarly threestructural metal sheets having a width of10cm and length of 90cm each were weldedtogether to form the base and side walls of

NIGERIAN JOURNAL OF TECHNOLOGY, VOL. 27 NO.2, SEPTEMBER 2008

48 OBASI, N.L. , AGUNWAMBA, J.C. AND EGBUNIWE, N.

the branch rectangular channel. Sevenbranch channels were constructed for eachchannel curvature and each of them was cutat the fitting end to be welded to the mainchannel in a manner that it represented oneof the off take angles to be investigated inthe study. The off take angle, θ was definedas the angle between the outer bank of themain channel and the inner bank of thebranching channel at the bifurcation point.According to this definition the suggestedoff take angles used in the study were10E,15E, 20E, 30E, 5E, 60E and 90E. Theupstream and downstream ends of the mainchannel were covered with 30cm by 30cmsheet metals by welding to retain the waterin the main channel during the experimentalwork. The downstream end was providedwith a spillway to allow excess water toflow out of the channel. At the bifurcationpoint a rectangular opening was created onthe main channel at the concave section ofthe model. A difference in invert levels of–15cm between the main and branching

channels was created to resemble a naturalsituation. The meandering section of themodel has an outer radius of 135cm andinner radius of 105cm indicating a mildbend that would not require futurechannelization.

The experimental materials consistedof the main channel, branch channels, twoelevated tanks, sediment supplier, pump,current meter kit, water, stop watch,measuring tape, beakers, plastic containers,electronic weighing balance, electric oven,Silver Nitrate, Sodium Chloride andPotassium Chromate. The experimental setup may be sub divided into three majorcomponents namely the water supply unit,the sediment supply unit and theregulatory/measuring unit. The circulationof water within the model was a closedsystem with a sediment trap provided atdownstream where the flow passed througha filter medium before being diverted backto the sump.

Fig. 1: Plan views ofmain channel,

off-take channel and crosssection.

Experimental procedureThe experiment was carried out at thehydraulic laboratory of the CivilEngineering Department, University ofNigeria, Nsukka in Enugu State of Nigeria.This study concentrated mainly on theinvestigation of the discharge distribution

patterns at concave channel bifurcation byvarying the off-take angles and the mainchannel flow rates. Four flow rates withcorresponding flow depths in the mainchannel of 16cm, 17cm, 18cm and 19cm

16 17 18 19and were represented by Q , Q , Q , Qwere used throughout the experimental



work. The flow depths corresponding toeach of the flow rates was established andmarked on the walls of the main channel asbench marks to ascertain each of the flowrates required The water in the sump waspumped to the first elevated tank thatsupplied water to the second overhead tank.The second overhead tank supplied waterdirectly to the model at controlled rates inline with the established points on the wallsof the main channel while the weir andspillway provided at the downstreammaintained the required water level in themain channel. The flow rates in the mainchannel were obtained by measuring themean velocities of the flow with the currentmeter apparatus at the marked points on thechannel walls and multiplying them withtheir corresponding wetted cross sectionalareas of the channel.

Flow distribution measurement atconcave channel bifurcation.Each of the seven off-take channelsrepresenting the recommended off-takeangles of 10E, 15E, 20E, 30E, 5E, 60E and90E was fixed at the opening created on theconcave side of the main channel undereach of the main channel flow rateconditions. Because the resulting off-takedischarges were low when compared withthe main channel flows, its dischargem e a s u r e m e n t w a s d e t e r m i n e dvolumetrically by collecting the flowdischarging freely into the air at the end ofthe channel in plastic containers at aspecified time. The volumes of thecollected water were measured with agraduated measuring cylinder and recordedagainst both the main channel flow ratesand the off-take angles as presented inFigure 2.

Dispersion experiment conducted at

concave channel bifurcation.The dispersion coefficient was determinedby carrying out tracer studies using theconstant distance variable time method.Twenty grams of Sodium Chloride (Nacl)was taken and dissolved in 200ml of water.For each of the specified main channel flowrates, a detention time was determined forthe concave channel bifurcation. For eachof the off-take angles the solution of thesodium chloride was introduced at thebeginning of the channel for every flow rateand samples of the tracer were collected atthe off-take channel and downstream of thebifurcation at regular time intervals of 20seconds. This was repeated for the sevenoff-take angles and four main channel flowrates. The samples collected were titratedwith Silver Nitrate solution after theaddition of Potassium Chromate. The datagenerated by the tracer experiment wereevaluated analytically to determine thevariance that is related to the dispersioncoefficient from the statistical momentmethod according to Levenspiel et al [14]thus;

1

2

According to Marecos de Monte and Mara[15]) the average flow time could bedetermined from the expression

3

Flow distribution pattern at concave



channel bifurcationFigure 2 shows the graphical representationof the specific discharge ratios and the off-take angles for concave channel bifurcation.Indications are that the off-take dischargesgradually increased with increase in off-take angles. It was also observed that anincrease in the parent channel dischargealso resulted in an increase in the off-takedischarge for any of the off-take anglesconsidered and the least off-take angleyielded minimum off-take discharge whilethe highest off-take angle yieldedmaximum off-take discharge. This isbecause for small off-take angles, the fluidparticles had the tendency to hit the rightwall of the off-take channel therebycreating a separation zone after thebifurcation point while some of the fluidparticles returned back to the main channel.Also, the sharp edged point at the

bifurcation point served as energydissipator thereby restricting the rate of thefluid motion to the off-take channel. It was,however, noticed that the width of theseparation zone increased with decrease inboth off-take angles and main channeldischarge. The knowledge of flowdistribution at channel bifurcation withrespect to the off-take angles is veryimportant in the selection and design ofwater intakes for irrigation and watersupply schemes particularly in the sizing ofthe off-take channel dimensions based onproper knowledge of the off-take discharge.

3 1The specific discharge ratios (q /q ) for theminimum and maximum discharges at theoff-take angle of 10 are 0.18 and 0.57o

respectively while for 90 off-take angleo

they are 1.19 and 2.51 respectively.

Fig. 2: Effect of off-take angles on specific discharge ratios for flows at concave channel bifurcation

Formulation of empirical expression.It has been proposed that the off-takedischarge could be proportional to the off-take angle (0), main channel flow (Q),dispersion coefficients for main and off-

1 3take channels (D , D ) and the Reynoldsnumber (Re) and may be expressed in theforms presented below:

4

0 2, 3 4, 5Where α , α α , α and α are constants.Linearlizing this equation by taking thesemi log and log of both sides of theequation gave the following expressionsthus:



5

6

7

Introducing a sine function in the aboveequation could lead to the modifiedexpression of the off-take discharge, thus:

8

Similarly, this expression can be linearlizedby taking the semi log and log of both sides

of the equation, thus:

9

10

11

The regression model constants weredetermined with the statistical package forthe social sciences (SPSS + PC) and theresults for each model equation arepresented in Tables 1 and 2.

Table 1: Predicting equation for flow distribution at concave channel bifurcation.

S/N Model equationRegression Coefficients Regression

correlationcoefficient R

Standard Error0 2 3 4 5α α α α α

1

-4.6

959

50

0.0

04

83

5

-0.1

051

47

-1.9

497

34

×1

0

33

9.2

79

52

4

0.98166 0.07821

2

0.0

29

79

1

7.4

71

97

6×

10-4

0.0

02

54

3

-0.0

03

50

1

0.0

05

79

2 0.98151 9.048931×10–5

3

1.20

51

68

-0.0

525

69

1.24

78

33

-0.3

933

55

1.05

73

87 0.99974 9.418394×10–3

Table 2: Modified equation for predicting flow distribution at concave channel bifurcation.



S/N Model equationRegression Coefficients Regression

correlationcoefficient R

Standard Error

0 2 3 4 5α α α α α

1

-4.4

30

35

0

01

22

25

00

0.1

08

935

1.6

39

30

4×

10-5

15

2.0

354

65 0.97277 0.09508

2

0.0

30

67

8

-2.4

32

17

×1

0-5

0.0

04

028

-0.0

04

415

0.0

04

200 0.97840 9.772729×10–5

3

1.1

52

662

0.0

09

816

1.1

42

449

3

-0.3

29

293

1.1

72

536 0.99974 9.753988×10–5

Discussion of ResultsThe results of the flow distribution wereused to evaluate the existing correlationbetween the off-take discharge and theassociated flow parameters for all theproposed expressions by the authors.

Tables 1 and 2 show that all theproposed expressions yielded highregression correlation coefficient valuesranging between 9.8151E-1 to 9.9974E-1with standard error values ranging from9.754E-5 to 7.821E-2. The high regressioncorrelation coefficient values recoded in theregression analysis suggests that there is ahigh degree of correlation between the off-take discharge and the proposed hydraulicparameters. Table 1 shows that the semi logexpression of the off-take dischargeindicates that the dispersion coefficients areinversely proportional to the off-takedischarge while the second equation ofsemi logs on the hydraulic parameters showthat the off-take discharge is inverselyproportional to only the Reynolds numbers.These two expressions have regressioncorrelation coefficient values of 9.8166E-1

and 9.8151E-1 with standard errors of7.821E-2 and 9.049E-5 respectively.However, the third expression with the logon both sides of the expression (Table 1) aswell, shows that the off-take discharge isinversely proportional to both the off-takeangle and the Reynolds numbers with aregression correlation coefficient value of9.9974E-1 and standard error value of9.4184E-5. The third expression with thehighest value of regression correlationcoefficient and the least standard errorvalue could be adopted as the modelpredicting equation for the off-takedischarge, thus:

12

13

Considering the expressions in Table 2, theresult of the semi log expression of the off-take discharge is theoretically in agreement



with the proposed expression stating thatthe off-take discharge is proportional to thehydraulic parameters except that it has thelowest regression correlation coefficientvalue of 9.7277E-1 and the highest standarderror value of 9.508E-2. The result of thesecond expression having semi log on thehydraulic parameters is similar to that ofthe model equation with log on both sidesof the expression indicating that the off-take discharge is inversely proportional tothe off-take angle and Reynolds numberswith the second highest regressioncorrelation coefficient value of 9.7840E-1and standard error value of 9.773E-5.Finally, the expression with the log on bothsides of the modified equation shows thatthe off-take discharge is inverselyproportional to the Reynolds numbers andhas the highest regression correlationcoefficient value of 9.9974E-1 with astandard error value of 9.754E-5. It can berightly suggested that the expression withthe highest regression correlationcoefficient and lowest standard error valuesmay predict fairly the off-take discharge forthe concave channel bifurcation. For themodified equation, it could be assumed thatthe off-take discharge is inverselyproportional to the Reynolds numbers asshown by the expression in Table 2, thus:

14

15

It can be generally seen from Tables 1 and2 that the sine function introduced in themodel equation has significant effect on the

proposed pattern of relationship betweenthe off-take discharge and the hydraulicparameters and its effect on the predictedvalues will be confirmed by the result of themodel verification. Both equationsproduced precise coefficients correlationvery close to unity, though the modelequation could be preferred due to its lowerstandard error value of 9.418394 ×10 as-3

against 9.753988 ×10 for modified-3

equation. Model Verification.Both model equations were verified withexperimental data at each specified off-takeangles. Figures 2 - 5 show the plots of theoff-take angles against the measured andpredicted off-take discharges for differentmain channel flows and the results showthat the off-take discharge graduallyincreases as the off-take angles increasewhich is in agreement with the result of theexperimental study. Considering the modelresults under the minimum main channel

16discharge of Q , the divergence in resultvalues for both model and laboratory testswas above 30% for the minimum off-take

angle of 10E while that of 45E off-takeangle was about 12%. The divergence ofresult values for the off-take dischargesobtained for the other off-take angles werewithin 20%. The predicted values for all theoff-take angles show a divergence in valuesless than 20% while only that of 10 showed0

a divergence of more than 30%. This couldbe attributed to the errors inherent inmeasurements carried out particularly in

16 17low flows as noticed in Q and Q . Themodel results for the main channel

17 discharge of Q showed also that thedivergence in results for 10E off-take angle



is about 30% while the results of the otheroff-take angles showed values less or equalto 20%. For the minimum main channel

16discharge (Q ), the off-take dischargevalues predicted by the model equationwere less than those obtained from thelaboratory test for off-take angles between10E and 30E with corresponding percentagedifference in values between 10 and 28. At45E off-take angle the predicted value wasequal to that obtained from the laboratorytest while the predicted valued for off-takeangles of between 60E and 90E were above

those obtained from the laboratory testswith a maximum of about 17%.

For the modified model equation, thepredicted values were mostly lower thanthose measured values except for the 90Eoff-take angle that was slightly above themeasured values by 2.3%. The trenddisplayed by the modified model equationwas that the degree of divergence washighest for the minimum off-take angle of10E which gradually decreased with

increase in off-take angles. It could be seenthat the 90E off-take angle showed theminimum divergence in values of about2.3%. Considering the main channel

17discharge of (Q ), the off-take dischargevalues predicted by the model equationwere less than the values obtained from thelaboratory tests for the off-take anglesbetween 10E and 20E while those for the

off-take angles between 30E and 90E weregreater than the laboratory measureddischarges. The highest percentagedivergence value of 24% was recorded for

the 10E and 90E off-take angles while theminimum percentage divergence value of4.2% was recorded for 20E off-take angle.The difference in values between themodified and model predicted values wereless than the measured laboratory valuesand were seen to be decreasing between 30to 2% for the off-take angles of 10E to 30Ewhile the predicted values for off-takeangles of 45E to 90E were higher than the

measured laboratory values. The minimumoff-take angle of 10E showed the highestpercentage divergence in values of 30 whilethe minimum percentage divergence of 20was recorded for the 45E off-take angle.Both model and modified equations did notshow any definite pattern in the variation ofpredicted and measured values.

ConclusionA comparative study of the various mainchannel flows with different off-take anglesshowed that as the off-take angles and mainchannel flow increased, the off-takedischarge equally increased. Thequantitative estimates of errors obtained inthe measured and predicted values is withinthe range of less than 2.4% to 15%indicating that the theoretical modelsdeveloped could reasonably be applied inthe determination of the off-take dischargesat concave channel bifurcations. This studyis useful for the determination of thedistribution flow pattern at concave channelbifurcation.



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