Prediction of zonal and total discharges in smooth straight prismaticcompound channels using regression modeling

Issam A. Al-Khatib a,n, Khaled A. Abaza b, Ismail Abu Fkhidah c

a Institute of Environmental and Water Studies, Birzeit University, P.O. Box 14, Birzeit, West Bank, Palestineb Department of Civil Engineering, Birzeit University, P.O. Box 14, Birzeit, West Bank, Palestinec Applied Statistics Master Program, Birzeit University, P.O. Box 14, Birzeit, West Bank, Palestine

a r t i c l e i n f o

Article history:Received 13 August 2013Received in revised form5 March 2014Accepted 11 May 2014Available online 18 May 2014

Keywords:Compound channelDischargeOpen channel flowRegression analysis

a b s t r a c t

Experimental results are presented concerning the zonal and total discharge distribution and character-istics in a compound channel cross-section comprising one rectangular main channel and twosymmetrical floodplains. The discharges in the main channel, floodplains, and total compound channelare found to be highly correlated to several dimensionless parameters defined using the compoundchannel cross-section geometry. Multi-variable regression analysis was utilized for developing predictivemodels that can estimate the main channel, floodplains and total discharges as a function of fourdifferent dimensionless parameters. The developed models to predict the zonal and total discharges incompound channels are found to be highly significant according to several major statistics including themodel standard error, coefficient of determination (R2), and F-statistic. The developed multi-variableregression-based models are also tested for validity using available experimental data. Several statisticaltests applicable to the analysis of residuals have indicated the effectiveness of the developed predictivemodels.

& 2014 Elsevier Ltd. All rights reserved.

1. Introduction

During the last centuries, many large urban communities havedeveloped on the river floodplains due to the demographicpressure and consequently increased utilization of rivers. Thishas led to the deaths of many people and to the increase ofeconomic costs when flooding occurs. In recent years, the disasterscaused by floods constitute about one-third of the losses caused bynatural disasters all over the world, which is the main cause ofmore than half of the deaths. Thorough analysis of the varioustrends of the damage caused by floods shows that these numbershave increased significantly in recent years [1].

There is a considerable range for the existence of errors inestimating the amount of water flow in the river channel whenconsidering a compound channel that consists of two or three zones:one or two side floodplains and a central main channel [2–6].

Usually compound channels are formed in natural rivers thatcarry most of the water at the bottom of the channel, and higherflows above them. This results in the reduction of channel erosionat lower flows and self-forming low flow channels that strikebanks [7,8]. Many practical problems in river engineering requireaccurate flow predictions in compound channels. Many practical

applications in hydraulic engineering also require accurate pre-dictions of flow in compound channels. It helps the practitionersin the development of vital information regarding flood protectionplans, building of hydraulic structures, the economic developmentof floodplain areas for parks and agriculture, and prediction ofsediment load so as to plan for effective preventive measures[1,5,9–11]. The water often flows in the main channel while thefloodplains are dry most of the time, but they are of particularimportance during flood events. Usually these floodplains extendlaterally away from the main channel and increase the transmis-sion capacity during flood events [12–14].

Strong lateral momentum exchange between the main channeland floodplains normally takes place across their interface surfaceaccording to the high gradient of flow velocity. This momentumtransfer between main channel and floodplains significantlyreduces the flow conveyance of compound channel [15–18]. Thiswas also emphasized by the experimental studies which con-cluded that momentum transfer slows down the flow in the mainchannel while accelerating the flow in the floodplains [19,20]. Thisresults in flow resistance and reduces the capacity of the com-pound channel [21–24]. When floodplains become submerged, thedifference in flow velocities between the main channel andfloodplains generates secondary currents and mixing patterns[25–27].

The estimation of discharge in compound channels with mainchannel and floodplains remains a major challenge to researchers

Contents lists available at ScienceDirect

journal homepage: www.elsevier.com/locate/flowmeasinst

Flow Measurement and Instrumentation

http://dx.doi.org/10.1016/j.flowmeasinst.2014.05.0070955-5986/& 2014 Elsevier Ltd. All rights reserved.

n Corresponding author. Fax: þ97 22 2982120.E-mail addresses: ikhatib@birzeit.edu, ikhatib2012@yahoo.com (I.A. Al-Khatib).

Flow Measurement and Instrumentation 38 (2014) 40–48

[28–30]. The works by different researchers also stressed theimportance of considering the main channel–floodplains interac-tion consequences [31–34].

Therefore, the need for accurate and simple methods toestimate the total and zonal discharges in compound channelcross-sections is consequently important. In this paper, an experi-mental study was carried out for the purpose of estimating thethree discharge components in the main channel, floodplains andtotal compound channel. The results of the experimental workwere used to develop multi-variable regression models that canpredict the three discharge components in symmetrical compoundchannels.

2. Experimental setup and experiments

A series of experiments was conducted at the HydromechanicsLaboratory of the Middle East Technical University, Ankara, Turkey,to measure the total discharge in a smooth straight prismaticcompound channel. The experiments were conducted using a glasswalled laboratory flume with 11.0 m length, 0.67 m width, 0.75 mdepth, and 0.005 bottom slope. In this study, it is assumed that thechannel bed slope is equal to the energy gradient slope. Arecirculation system of water supply is established, in which wateris pumped from an underground sump to an overhead tank fromwhere water can flow to the flume; it passes through theexperimental channel under gravity and is allowed to flow overa rectangular sharp crested suppressed weir mounted in the inletbox of the flume. From the volumetric tank, water flows back tothe underground sump. Head measurements over the crest of theweir were done by a point gage of 0.01 cm accuracy and apredetermined calibration curve of the weir was used to deter-mine the discharges. The point gauge was used along the center-line of the flume for head measurements. All depth measurementswere done with respect to the bottom elevation of the flume. Themaximum capacity of the system was around 110 l/s.

Six prismatic Plexiglass models having rectangular compoundchannel cross-section with two equal floodplains, two differentstep heights (Z) and three main channel widths (B) were tested. Asymmetrical cross-section with a center channel section was thuscreated. The ratio of the overall channel width (Bo) to the width ofthe main channel (B) thus ranged from 1.49 to 3.35 as provided inTable 1. The ratio (Bo/B) was varied by adding adjustable side wallsto each of the flood plains in pairs to give a symmetrical cross-section as required.

The models were placed at about mid-length of the laboratoryflume. Fig. 1 shows the plan view and cross-section of the typicalmodel with symbols designating important dimensions of the

model elements. A total of 6 model combinations were testedusing 3 different main channel widths (B) and 2 different stepheights (Z). Table 1 provides the values of all channel dimensionsused in the 6 tested models along with several dimensionlessparameters to be used in the analysis of experimental results andin the development of multi-variable regression-based predictionmodels. In this study, the tested model types are denoted by BiZj(i¼20, 30, 45; j¼5, 10). The subscripts i and j designate thenumerical values of (B) and (Z) in centimeter used in this study,respectively. The required experiments were first conducted in themodels with the smallest B (¼20 cm) and varying Z values (¼5 cm and 10) and then B was increased to 30 cm at the requiredamount of Z (¼ 5 cm and 10), and finally for B¼45 cm with thesame two values of Z. The entrance angles (θ and β) were 26.5651and 153.351, respectively. The transition length was twice thefloodplain width (Bf).

In order to determine the velocity distribution in the rectan-gular compound cross-sections, the channel cross-section wasdivided into a number of successive lines normal to the directionof the flow. Then, the total and static heads were measured atseveral points along these normal lines by the use of a pitot(Preston) tube with an external diameter of 7 mm. Additionalpoints were taken close to the channel boundary while thedistances between the points were increased towards the freesurface. The velocity area method was used to find the dischargefor each zone of the cross-section, which could then be summedup to give the full cross-sectional discharge in all models.

3. Presentation and discussion of results

In this section, the impact of several channel cross-sectiongeometric parameters on the three discharge components willbe investigated for the purpose of identifying the potentialparameters to be included in the development of the proposedmulti-variable regression models. These cross-section geometricparameters include the relative depth (Yr), mean channel width (B)and step height (Z), and the dimensionless ratio yf/Z.

3.1. Variation of discharge with yf/h

For any prediction model to be adequate, it must accuratelydescribe not only the total cross-sectional discharge (QT) but alsothe main channel and floodplain discharges Qmc and Qf, respec-tively. The average discharges in the main channel (Qmc), flood-plains (Qf), and total channel cross-section (QT) for the six differentmodels, B20Z5, B20Z10, B30Z5, B30Z10, B45Z5 and B45Z10, inorder, are shown in Fig. 2–7. These discharges are depicted in

Nomenclature

The following symbols are used in this paper:

B bottom width of the main channelBf floodplain channel widthBO bottom width of the upstream channelCV coefficient of variationh main channel water depthLe entrance channelLapp approach channelln natural logarithm functionMSE mean squared errorsMSPR mean of the squared prediction errorsQmc mean main channel volumetric flow rate

Qf mean floodplain volumetric flow rateQT mean full cross-sectional volumetric flow rateR2 coefficient of determinationR1 Bf/ZR2 Bo/BR3 yf/ZR4 yf/hVIF variance inflation factorYr independent variable representing relative depth ¼yf/

hyf floodplain water depthZ step heightθ1 and θ2entrance angles

I.A. Al-Khatib et al. / Flow Measurement and Instrumentation 38 (2014) 40–48 41

relation to the relative depth (Yr) because it reflects the effect ofthe cross-section geometry on the discharge distribution in thecompound channel. The relative depth (Yr) is defined as the ratioof the floodplain water depth to the main channel water depth(yf/h) , which is a dimensionless quantity.

From Figs. 2–7, as a general trend, one can conclude that thevalues of Qmc, Qf, and QT increase as relative depth increases. Also,there exists a point of intersection between the main channel flow

(Qmc) and floodplain flow (Qf) considering the models B20Z5 andB20Z10. This means that at these two points of intersection, thedischarges are equal in both the main channel and the floodplain.The corresponding (Yr) values for these two points of intersectionare about 0.57 and 0.55 for models B20Z5 and B20Z10, respec-tively. Before the point of intersection Qmc is consistently greaterthan Qf for these two models, while after the point of intersectionQf becomes greater than Qmc. This trend can take place only if themain channel width, B, is smaller than half of the total channelwidth, BO, indicating that it is possible for the floodplain dischargeto become larger than the main channel discharge provided thewater depth is sufficiently large. However, for models, B30Z5 andB30Z10, the water depth was not adequately high to depict this

Table 1Dimensions and dimensionless values of tested models.

Model B (cm) Z (cm) Bf (cm) Le L BO (cm) θ (deg) Β (deg) BO/Bf (–) BO/Z (–) BO/B (–) Bf/Z (–) Bf/B (–) B/Z (–)

B20Z5 20 5 23.5 47 294 67 26.57 153.43 2.85 13.40 3.35 4.70 1.18 4.00B20Z10 20 10 23.5 47 294 67 26.57 153.43 2.85 6.70 3.35 2.35 1.18 2.00B30Z5 30 5 18.5 37 294 67 26.57 153.43 3.62 13.40 2.23 3.70 0.62 6.00B30Z10 30 10 18.5 37 294 67 26.57 153.43 3.62 6.70 2.23 1.85 0.62 3.00B45Z5 45 5 11 22 294 67 26.57 153.43 6.09 13.40 1.49 2.20 0.24 9.00B45Z10 45 10 11 22 294 67 26.57 153.43 6.09 6.70 1.49 1.10 0.24 4.50

Plan View

Section 1-1

Fig. 1. Definition sketch of the flume used in the experiments.

Qmc = 0.287Yr2 - 0.227Yr + 0.051

R² = 0.990Qf= 0.451Yr

2 - 0.360Yr + 0.074R² = 0.989

QT =0.739 Yr2-0.587 Yr +0.126

R² = 0.989

0

0.02

0.04

0.06

0.08

0.1

0.12

0.30 0.40 0.50 0.60 0.70 0.80

Mea

n Fl

ow (m

3/s

ec)

Yr

QmcQfQT

Fig. 2. Variation of mean flow components versus Yr for compound cross-sectionB20Z5.

Qmc= 0.208Yr2 - 0.056Yr + 0.016

R² = 0.991Qf= 0.351Yr

2 - 0.126Yr + 0.013R² = 0.992

QT= 0.560Yr2 - 0.182Yr + 0.030

R² = 0.992

0

0.02

0.04

0.06

0.08

0.1

0.12

0.10 0.20 0.30 0.40 0.50 0.60

Mea

n Fl

ow (m

3 /sec

)

Yr

Qmc

Qf

QT

Fig. 3. Variation of mean flow components versus Yr for compound cross-sectionB20Z10.

Qmc = 0.325Yr2 - 0.209Yr + 0.042

R² = 0.992Qf= 0.240Yr - 0.161Yr + 0.028

R² = 0.991QT = 0.565Yr

2 - 0.370Yr + 0.071R² = 0.992

0

0.02

0.04

0.06

0.08

0.1

0.12

0.200 0.400 0.600 0.800

Mea

n Fl

ow (m

3 /sec

)

Yr

Qmc

Qf

QT

Fig. 4. Variation of mean flow components versus Yr for compound cross-sectionB30Z5.

I.A. Al-Khatib et al. / Flow Measurement and Instrumentation 38 (2014) 40–4842

point of intersection. A point of intersection is not possible formodels B45Z5 and B45Z10 since the main channel width is largerthan half of the total channel width.

Figs. 2–7 also provide for each discharge component a 2nddegree polynomial that relates the discharge to relative depth.These 2nd degree models represent the best-fit curves essentiallygenerated to minimize the squared sum of residuals. These models

are associated with very high coefficient of determination (R2)ranging from 0.989 to 0.998, indicating their high predictivestrength. These models can be used to predict the three dischargecomponents using the relative depth as the sole independentvariable.

3.2. Effect of B and Z on discharge as a function of Yr

In order to investigate the effect of the main channel width (B)and step height (Z) on the values of the discharge components, thevariations of Qmc, Qf, and QT with Yr were plotted separately for allmodels tested and are shown in Figs. 8, 9 and 10, respectively.From these figures, it can be stated that there are clear effects ofthe main channel width and step height on the discharge compo-nents. For a given Yr, the value of Qmc has increased as the mainchannel width and step height have increased. The case is differentfor the variation of Qf with Yr; the Qf value has increased withdecreasing main channel width and it has increased with increas-ing step height considering a given Yr value. For models with afixed main channel width, if the step height is increased ordecreased, the (Qf) or (Qmc) values significantly change for smallvariations in the (Yr) value. Fig. 10 shows that the effects of B and Z

Qmc = 0.256Yr2 - 0.052Yr + 0.024

R² = 0.998Qf = 0.192Yr

2 - 0.056Yr + 0.005R² = 0.997

QT = 0.448Yr2 - 0.109Yr + 0.03

R² = 0.998

0

0.02

0.04

0.06

0.08

0.1

0.12

0.10 0.20 0.30 0.40 0.50 0.60

Mea

n Fl

ow (m

3 /sec

)

Yr

Qmc

Qf

QT

Fig. 5. Variation of mean flow components versus Yr for compound cross-sectionB30Z10.

Qmc = 0.385Yr2 - 0.222Yr + 0.046

R² = 0.989Qf = 0.089Yrx2 - 0.053Yr+ 0.008

R² = 0.992

QT = 0.475Yr2 - 0.275Yr + 0.055

R² = 0.99

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.20 0.40 0.60 0.80

Mea

n Fl

ow (m

3 /sec

)

Yr

Qmc

Qf

QT

Fig. 6. Variation of mean flow components versus Yr for compound cross-sectionB45Z5.

Qmc= 0.300x2 - 0.025x + 0.037R² = 0.997

Qf = 0.076Yr2 - 0.017Yr + 0.002

R² = 0.998QT = 0.377Yr

2 - 0.042Yr + 0.04R² = 0.997

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.10 0.20 0.30 0.40 0.50 0.60

Mea

n Fl

ow (m

3 /sec

)

Yr

Qmc

Qf

QT

Fig. 7. Variation of mean flow components versus Yr for compound cross-sectionB45Z10.

0

0.02

0.04

0.06

0.08

0.1

0.12

0.0 0.2 0.4 0.6 0.8

Mea

n Fl

ow Q

mc

(m3 /s

ec)

Yr

B20Z5B20Z10B30Z5B30Z10B45Z5B45Z10

Fig. 8. Variation of Qmc versus Yr for the six tested compound cross-sections.

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.0 0.2 0.4 0.6 0.8

Mea

n Fl

ow Q

f(m

3 /sec

)

Yr

B20Z5B20Z10B30Z5B30Z10B45Z5B45Z10

Fig. 9. Variation of Qf versus Yr for the six tested compound cross-sections.

I.A. Al-Khatib et al. / Flow Measurement and Instrumentation 38 (2014) 40–48 43

on the total discharge (QT) are similar to their effects on the mainchannel discharge (Qmc).

3.3. Variation of discharge with yf/Z

The experimental results associated with the three dischargecomponents (Qmc, Qf, and QT) are shown in Figs. 11–16 as afunction of the yf/Z ratio for the six models tested. It can be notedthat the three discharge components are directly proportional tothe yf/Z ratio as expected. A best-fit curve has been generated foreach discharge component as a function of the (yf/Z) ratio.A second degree polynomial model has been also developed foreach discharge component as presented in these figures with a veryhigh coefficient of determination (R2). Figs. 11 and 12 show a trendsimilar to that depicted in Figs. 2 and 3 wherein an intersectionpoint exists between the main channel and floodplain discharges.

3.4. Effect of B and Z on discharge as a function of yf/Z

Fig. 17 depicts the discharge in the main channel (Qmc) versusthe yf/Z ratio for the six tested models. It can be noted that thisdischarge component, for a specific yf/Z ratio, is directly propor-tional to the main channel bottom width (B) considering aconstant step height (Z). It is also directly proportional to the step

height (Z) considering a specific yf/Z ratio and constant B value aswould be expected. On the other hand, Fig. 18 shows that thedischarge in the floodplains (Qf) is inversely proportional to themain channel bottom width (B) considering a specific yf/Z ratioand constant step height (Z). However, the Qf is directly propor-tional to the step height (Z) for a specific yf/Z ratio and constant B

0

0.02

0.04

0.06

0.08

0.1

0.12

0.0 0.2 0.4 0.6 0.8

Mea

n Fl

ow Q

(m3 /s

ec)

Yr

B20Z5B20Z10B30Z5B30Z10B45Z5B45Z10

Fig. 10. Variation of QT versus Yr for the six tested compound cross-sections.

Qmc = 0.009(yf /z) 2 + 0.021(yf /z) + 0.008R² = 0.994

Qf = 0.024(yf /z) 2 + 0.014(yf /z) - 0.001R² = 0.997

Q = 0.033(yf /z) 2 + 0.036(yf /z) + 0.006R² = 0.996

0

0.02

0.04

0.06

0.08

0.1

0.12

0.0 0.5 1.0 1.5

Mea

n Fl

ow (m

3 /sec

)

yf /Z

Qmc

Qf

QT

Fig. 11. Variation of mean flow components versus yf/Z for compound cross-sectionB20Z5.

Qmc = 0.009(yf /z) 2 + 0.021(yf /z) + 0.008R² = 0.994

Qf = 0.024(yf /z) 2 + 0.014(yf /z) - 0.001R² = 0.997

Q = 0.033(yf /z) 2 + 0.036(yf /z) + 0.006R² = 0.996

0

0.02

0.04

0.06

0.08

0.1

0.12

0.0 0.5 1.0 1.5

Mea

n Fl

ow (m

3 /sec

)

Yf /Z

Qmc

Qf

QT

Fig. 12. Variation of mean flow components versus (yf/Z for compound cross-section B20Z10.

Qmc = 0.002(yf /z) 2 + 0.015(yf /z) + 0.002R² = 0.998

Qf = 0.002(yf /z) 2 + 0.009(yf /z) - 0.002R² = 0.998

QT = 0.005(yf /z) 2 + 0.024(yf /z)R² = 0.998

0

0.02

0.04

0.06

0.08

0.1

0.12

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Mea

n Fl

ow (m

3 /sec

)

yf /Z

Qmc

Qf

QT

Fig. 13. Variation of mean flow components versus yf/Z for compound cross-sectionB30Z5.

Qmc = 0.009(yf /z) 2 + 0.036(yf /z) + 0.015R² = 0.999

Qf = 0.011(yf /z) 2 + 0.015(yf /z) - 0.001R² = 0.999

QT = 0.020(yf /z) 2 + 0.052(yf /z) + 0.013R² = 0.999

0

0.02

0.04

0.06

0.08

0.1

0.12

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

Mea

n Fl

ow (m

3 /sec

)

yf /Z

Qmc

Qf

QT

Fig. 14. Variation of mean flow components versus yf/Z for compound cross-sectionB30Z10.

I.A. Al-Khatib et al. / Flow Measurement and Instrumentation 38 (2014) 40–4844

value. Fig. 19 shows the total compound channel discharge (QT)with a trend similar to that associated with the main channeldischarge as depicted in Fig. 17.

4. Development of multi-variable regression models

Several geometric dimensionless parameters expected to influ-ence the three discharge components have been investigated. Themost important parameters that have been found to significantlyaffect the three discharge components are those related to thecompound channel cross-section dimensions. Multi-variable linearregression analysis has been used to identify the geometricdimensionless parameters that have the greatest influence onthe three discharge values. This resulted in the identification offour dimensionless ratios called R1–R4 as defined in Eqs. (1)–(3),where R1¼Bf/Z, R2¼Bo/B, R3¼yf/Z, and R4¼yf/h¼Yr.

Eqs. (1)–(3) are developed to relate the three discharge com-ponents, namely, Qmc, Qf, and QT to the four dimensionless ratiosR1–R4. The three developed models are non-linear in form but theyshould first be transformed to linear form before applying theregression analysis. These regression models can be used to

Qmc = 0.003(yf /z) 2 + 0.024(yf /z) + 0.005R² = 0.997

Qf = 0.000(yf /z) 2 + 0.005(yf /z) - 0.001R² = 0.997

QT = 0.004(yf /z) 2 + 0.029(yf /z) + 0.003R² = 0.997

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Mea

n Fl

ow (m

3 /sec

)

yf /Z

Qmc

Qf

QT

Fig. 15. Variation of mean flow components versus yf/Z for compound cross-sectionB45Z5.

Qmc = 0.008(yf /z) 2 + 0.064(yf /z) + 0.028R² = 0.998

Qf = 0.004(yf /z) 2 + 0.009(yf /z)R² = 0.999

QT = 0.013(yf /z) 2 + 0.073(yf /z) + 0.027R² = 0.999

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.0 0.2 0.4 0.6 0.8 1.0 1.2

Mea

n Fl

ow (m

3 /sec

)

yf /Z

Qmc

Qf

QT

Fig. 16. Variation of mean flow components versus yf/Z for compound cross-sectionB45Z10.

0

0.02

0.04

0.06

0.08

0.1

0.12

0.0 1.0 2.0 3.0 4.0

Mea

n Fl

ow Q

mc

(m3 /s

ec)

yf /Z

B20Z5B20Z10B30Z5B30Z10B45Z5B45Z10

Fig. 17. Variation of Qmc versus yf/Z for the six tested compound cross-sections.

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.0 1.0 2.0 3.0 4.0

Mea

n Fl

ow Q

f(m

3 /sec

)

yf /Z

Fig. 18. Variation of Qf versus yf/Z for the six tested compound cross-sections.

0

0.02

0.04

0.06

0.08

0.1

0.12

0.0 1.0 2.0 3.0 4.0

Mea

n Fl

ow Q

T(m

3 /sec

)

yf /Z

Fig. 19. Variation of QT versus yf/Z for the six tested compound cross-sections.

I.A. Al-Khatib et al. / Flow Measurement and Instrumentation 38 (2014) 40–48 45

predict the three discharge components provided that they areused in conditions similar to those associated with their develop-ment.

ln Qmc ¼ �7:274þ 7:238ðR1Þ0:5

þ3:965ðR4Þ2�0:001ðR2Þ5

�2:168ðR2Þ3

�0:029R3

ð1Þ

ln Qf ¼ �2:616�2:444ðR2Þ3

�1:526ðR1Þ0:5

þ0:491ðR4ÞðR2ÞðR3Þþ2:138 sin ½ðR4ÞðR2Þ�� �� 0:486

½ðR4ÞðR2Þ�ð2Þ

ln QT ¼ �3:946þ 2:630ðR1Þ0:5

�0:903R2

�120:349ðR4Þ15

�0:093½ðR1ÞðR2Þ�þ0:563½ðR2ÞðR3Þ� ð3Þ

It is to be emphasized that models with different combinations ofseveral geometric-based dimensionless parameters (ratios),including single parameter models, have been investigated in theprocess of developing the presented best-fit models. The best-fitmodels are the models incorporating the four outlined dimension-less parameters (R1–R4) as presented in Eqs. (1)–(3) and areassociated with the best predictive strength. Therefore, the pre-sented best-fit models contain the minimum number of indepen-dent parameters with deployed statistics indicating theirpredictive superiority over other investigated models.

The significance of the developed regression models has beensubjected to several statistical tests. The first group of testsexamines the reliability and strength of each model as a whole.This group includes the model R2 value, model standard error, andmodel F-statistic. Table 2 provides the values of these threestatistics for each predicted variable. It can be noticed that theR2 is very high, the standard error is considerably low, and theF-statistic is highly significant at 99.9%. The second group ofstatistics examines the significance of the model individual coeffi-cients, which includes the coefficient t-statistic and varianceinflation factor (VIF). Table 2 provides the model coefficients alongwith their relevant statistics. It is clear that all model coefficientsare highly reliable at 99.9% confidence level. The VIF coefficient

values are all below the recommended maximum tolerancevalue of 10.

The third group of tests has been performed to analyze theresiduals associated with the developed predictive regressionsmodels. The first test compares the mean of the squared predictederrors (MSPE) to the corresponding mean squared errors (MSE).Table 3 provides for each model the corresponding MSPE and MSEvalues. It can be noticed that the MSPE values are either less thanor equal to the corresponding MSE values, an indication that theresiduals associated with the tested sample of experimental dataare within the acceptable range. The second test includes thedevelopment of normal probability plots (Fig. 20 as an example),standardized residual histograms (Fig. 21 as an example), andstandardized residual scatter plots (Fig. 22 as an example). Fig. 20shows that the observed cumulative normal probabilities are veryclose to their corresponding expected cumulative probabilities.Fig. 21 indicates that the standardized residuals generally adhereto the assumption of normal probability. Fig. 22 provides scatterplots of the standardized residuals indicating that the standar-dized residuals are highly independent. Therefore, the performedresidual analysis provides a good validation of the developedmulti-variable regression models to be used in predicting thethree discharge components.

As outlined previously, this study has been concerned withsmooth straight prismatic compound channels. Roughness coeffi-cient has a significant impact on flow characteristics. For example,Al-Khatib et al. conducted two sets of experiments with smoothand rough surfaces considering both main channel and floodplainin asymmetric straight compound channels [11]. It was observedthat in the case of rough flume, the differences among the meanvelocities in the main channel, floodplain and the whole com-pound cross-section are substantially higher compared to thesmooth case. On the other hand, the depicted velocity values arehigher in the case of smooth flumes as would be expected. It was

Table 2Summary of statistics of new multiple regression predictive models.

Predicted variable Model R2 Model standard error Model F-statistic Model coefficients Coefficient t-statistic Confidence level (%) VIF coefficient

ln Qmc 0.992 0.09127 931.644 – – 99.9 –

�7.274 �86.099 99.97.238 49.039 99.9 2.3663.965 37.614 99.9 2.871�0.001 �9.958 99.9 2.248�2.168 �14.219 99.9 3.090�0.029 �2.472 99.4 2.451

ln Qf 0.990 0.16464 798.084 � � 99.9 ��2.616 �9.643 99.9 ��2.444 �11.699 99.9 1.782�1.526 �28.029 99.9 2.3610.491 29.766 99.9 2.7362.138 10.958 99.9 4.944�0.486 �7.179 99.9 7.383

ln QT 0.981 0.09309 528.035 – – – –

�3.946 �21.301 99.9 –

2.630 11.410 99.9 3.608�0.903 �5.967 99.9 2.761�120.349 �14.204 99.9 3.300�0.093 �19.075 99.9 9.2670.563 41.369 99.9 5.565

Table 3MSE and MSPR associated with the multiple-variable regression models.

Dependent variable MSE MSPR

ln Qmc 0.008 0.010ln Qf 0.021 0.027ln QT 0.009 0.011

I.A. Al-Khatib et al. / Flow Measurement and Instrumentation 38 (2014) 40–4846

also observed that the relative depth values associated with therough flume are higher than the corresponding values associatedwith the smooth flume. In natural rivers the roughness coefficient

depends on several factors, including unsteadiness characteristics,surface roughness, vegetation around the section, and channelirregularity, and the exact roughness values are often uncertain[35]. Therefore, in the case of natural rivers, the authors recom-mend that a test channel model be constructed with roughnesscharacteristics similar to those of the actual river case and then theprocedure followed in this paper can be used to develop applicableregression models.

5. Conclusion

The experimental results for three discharge components havebeen presented for a symmetric rectangular compound channelwith 6 different cross-section geometries. The experimentalresults have been analyzed with respect to several dimensionlessparameters derived mainly from the channel cross-section geo-metry. It has been found that the three discharge components(Qmc, Qf, and QT) are directly proportional to four channel cross-section dimensions, namely, the relative depth ratio (yf/h), mainchannel width (B), step height (Z), and the water depth ratio (yf/Z).The best-fit models, generated using solely the relative depth ratioor the water depth ratio (yf/Z), have indicated their high strengthin predicting the three discharge components.

The experimental discharge data associated with the 6 testedcompound channel cross-sections have been used to developmulti-variable predictive regression models. A non-linear multi-variable regression model has been generated for each dischargecomponent using the four dimensionless ratios, namely, Bf/Z, Bo/B,yf/Z, and yf/Z. The developed three multi-variable regressionmodels have been subjected to several statistical tests to examineand validate their predictive strength. All deployed statistical testshave indicated that the developed regression models are highlyreliable in predicting the three discharge components using onlythe compound channel cross-section dimensions. However, it is tobe emphasized that regression models are typically applicable toconditions similar to those used in their development.

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