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B.R. Andharia, et al. / (IJAEST) INTERNATIONAL JOURNAL OF ADVANCED ENGINEERING SCIENCES AND TECHNOLOGIES Vol No. 4, Issue No. 2, 036 - 041

Mathematical Model Based on Mobility Index for the Prediction of Flow Resistance (Case study of Savkheda Gauging Station Using Tapi River Data, India) B.K.Samtani2

B.R. Andharia*1

Prof. & Head, Civil Engineering Department, Sardar Vallabhbhai National Institute of Technology, Surat, Gujarat - 395 007, India (E-mail:samtanibk@yahoo.com)

Ph.D. Research Scholar, Civil Engineering Department, Sardar Vallabhbhai National Institute of Technology, Surat, Gujarat - 395 007, India. (Corresponding author, E-mail: andharia.bhoomi@gmail.com )

II. OBJECTIVES The main objectives of this paper are: 

To compute the resistance to flow in alluvial channel using various methods based on various approaches,



To establish relation between the non-dimensional number Mobility Index and friction factor obtained by different methods.



To built the mathematical model to predict the friction factor using Mobility Index.

Abstract - The friction factor of an open channel flow is generally affected by the characteristics of the fluid and flow, geometry of the channel and the characteristics of the channel boundary. Large research has been carried out for the study of friction factor using one or more parameters like Froude number, Relative flow depth, hydraulic mean depth, slope, flow Reynolds number, size of the particles etc. However, in each study only limited numbers of the parameters have been considered. In alluvial rivers depending upon the flow conditions, any of the above parameters may vary and predominate to change the friction factor. So, here the attempt has been made to compute the friction factor following the different methods and approaches based on the field data of 15 years of Tapi River for monsoon season and study the variation of fiction factor with Mobility Index. The mathematical model to predict the friction factor for the Tapi River at Savkheda Gauging station has been developed by using the average value of the friction factor obtained by various methods based on various approaches and the multiple regression analysis. The multiple regression analysis has been carried out by using the non-linear curve fitter of the origin 7.5 software. The mathematical model has been developed relating the Mobility Index with the friction factor.

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Keywords -Friction factor, Mobility Index, multiple regression analysis, mathematical model, Origin 7.5.

INTRODUCTION

Savkheda is one of the gauging stations on river Tapi. In the present paper, the last 15 year data collected from this gauging station is used to compute resistance to flow in terms of friction factor. The friction factor is computed for monsoon season. There are number of approaches used to compute friction factor but in this paper three approaches are used to compute friction factor. The field data of 15 years has been analyzed and computer programming in Ms-excel and origin has been used to carry out analysis of the data. The relationship between friction factor and Mobility Index has been established. The graphs are plotted for the above parameters using origin software and on the obtained results, statistical analysis is carried out. The Mobility Index prediction may help in assessing resistance to the flow in alluvial channel. The entire analysis is based on average diameters of particles. 1

Ms. Bhoomi Andharia, Ph.D. Research Scholar, Civil Engg.Dept, S.V.N.I.T., Surat, Gujarat395 007, India.E-mail ID: andharia.bhoomi@gmail.com, Tel: +91-261 201654, (M) +9109428489210, +91-09824467655,Fax No. +91-0261-2227334, 2228394

ISSN: 2230-7818

A. Study Area and Data Collection Tapi is the second largest westward flowing river of peninsular India. Total length of the river is 724 kilometers from origin to Arabian Sea. The Tapi basin (Fig.1) is situated between latitudes 20o N to 22o N, 80% of the basin lies in Maharashtra and the balance in the state of Madhya Pradesh and Gujarat. Central Water Commission, Tapi Division, Surat is regularly collecting daily data of discharge and sediment at gauging site Savkheda on river Tapi (Fig.2). Savkheda is situated at a distance of about 399 kilometers from origin. The daily data like discharge, area, bed slope, velocity, wetted perimeter, hydraulic mean depth, Manning‟s and Chezy‟s constants, average diameter of sediment, mean diameter of sediment etc. during monsoon are collected for 15 years period from 1981 to 1995 and 2000-2005 for study from the central water commission water year books (1980-1995, 2000- 2005). B. The Mobility Index The Mobility Index (MI) is related to the shear velocity and fall valocity of the sediment particle. MI 

mean shear velocity (m/sec) u *  fall velocity (m/sec) w

(1)

This dimensionless number is useful to help predict bed forms and it is encountered in the resistance formulas for the prediction of the height of the bed-forms. C. Friction Factor From the Darcy-Weisbach factor f can be deﬁned as

relation,

the

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friction

B.R. Andharia, et al. / (IJAEST) INTERNATIONAL JOURNAL OF ADVANCED ENGINEERING SCIENCES AND TECHNOLOGIES Vol No. 4, Issue No. 2, 036 - 041

8 gRS f u



8u* u  u 2 and u*

8 f

(2) Equation (2) reveals that the friction factor is the value of the velocity of fluid motion generated by shear stress. When the effects of form roughness are absent, the main source of friction factor is the skin roughness which is expressed through the grain Reynolds number u* R /  [discussed in detail by Wilcox et al. (2006) who has been working on these two parameters by using various laboratory flume runs]. This parameter depends on the Shields parameter; therefore, one needs to take into account the effect of τ* on the friction factor. Parameter τ* is non-dimensionalized using the critical value denoted as τ*c. The value of the critical Shields parameter was determined by Lamb et al. (2008) as  *C  0.15 S 0.25 (3) Equation (3) was obtained by an empirical fit to the data for slopes less than about 10%. The prediction of flow resistance in alluvial channels is needed for two major purposes: 

The estimation of stage discharge relationship

For the different set of the regimes identified using eq.6 (a), (b), (c) as Lower Regime or subcritical flow: F  1

(6a)

Transition Regime or critical flow: F  1

(6b)

Upper Regime or Super critical flow: F  1

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III. COMPUTATION OF THE FRICTION FACTOR The estimation of friction factor has been done by using the three methods. These methods have been selected as they are based on different approaches and all the important variables affecting the resistance of the flow are included. In order to develop the flow resistance equation for the Tapi river, average of the all the 3 methods have been utilized and then the multiple regression analysis has been carried out for the developing the final equation of resistance to flow using statistical analysis tools of Microcal Origin 7.5 software. A. Model I. Keulegan’s Equation (f1) [7]

For the hydraulic smoothness region, the distribution of the cross-sectional mean velocity in the form of friction factor f is v  u*

uR 8  5.75 lg *  3.25  f

(4)

For the hydraulic roughness region, the distribution of the cross-sectional mean velocity is v  u*

8 R  5.75 lg  6.25 f  R2 =0.784

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(5)

(6c)

Using the Variables from the Manning‟s equation Multiple Regression Analysis is done using the Origin 7.5 Software. For the Multiple Regression Analysis to fining out friction factor the variables used are discharge Q, Cross sectional area, Manning‟s n, hydraulic mean depth and slope. Here, Multiple Regression Analysis by considering the variables like A, R, S, and n. Equation (7) is used for computing modified „Q‟ Q = -364.89 + 1.96*A + 153.93*n - 40.54*R + 95265.31*S R2=0.98193 (7) Manning‟s n is found out from the Manning‟s equation using the modified Q from above (7). The friction factor f can be calculated from the modified Manning‟s n using relation [6]

 The estimation of sediment transport from the hydraulic characteristics of the channel by means of transport formula. Moreover, knowledge of the resistance characteristics of alluvial streams is of great value when dealing with the location of bridges, training works, flood control works, navigation and channel improvement, backwater computation due to confluences and barrages, mathematical and physical modelling of flow, prediction of aggradations and degradation due to presence of hydraulic structures and so on.

B. Model II. Modified Manning’s equation based on the method of multiple regression analysis by considering the variables like A, R, S, n etc. (f2)

f 

u R1 / 6  u* n g

8 f

(8)

C. Model III. Relationship based on multiple regressions analysis using factors effecting sediment transportation (f3) The Shields parameter τ* can be expressed as * 

u* 2    S   gd  S   gd

(9)

The shear stress combined linearly with the Froude number has already been considered using limited data [Colosimo et al. 1986; Afzalimehr and Anctil 1998Ref.1,2,5]. In the ranges of (0.01< τ*c <0.05) and slope (0.00081< S <0.01), using the  h   

relative flow depth  d 50  ,the ratio of Shields parameter to its  *   

critical value    , the Froude number (Fr) and bed slope (S) in the universal velocity distribution law, the friction factor was expressed from regression analysis as *C

 8 u 1  h    18.56  0.0058 *  54.28Fr  22342.12S   0.57 In  *C f u* k  d 50 

R 2  0.96

D. Relationship Parameters

(10) Based

Estimated

and

Computed

The method used for estimation of friction factor depends upon many variables. The seasonal average values of these parameters calculated by all the three methods are based on different approaches. It is observed that all the three methods used for the study involves all-important hydraulic parameters

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and sediment parameters. Hence, the comparisons of friction factor calculated for developing the resistance to the flow of this river are compatible. Looking to the above fact average of three methods can be used as base without any loss of accuracy. Therefore, average values of friction factor, estimated by three methods is considered as a reliable base for the comparison of data collected and relation between friction factor and Mobility Index are established. IV.

RESULTS

A multiple regression analysis is carried out between the measured and calculated basic data, viz., bed width discharge per unit width, flow area, hydraulic mean depth, velocity of flow, bed slope, avg. diameter of sediments with calculated average values of friction factor by above three approaches using statistical analysis tool of Origin 7.5 software by using non-linear curve fitter. Finally, (11) is derived for Savkheda gauging station of Tapi river. fMR= Co + C1 Q + C2 A +C3S + C4Bw +C5 V + C6R + C7davg+ C8u* + C9w (11) In equation (11), C, C1 , C2 , C3 , C4 , C5 , C6 , C7 ,C8, C9 are multiplying constants for Q,A,S,Bw,V,R,davg, u*and w are shown in Table I.

The value of friction factor obtained by various methods and their variations with the Mobility Index are shown in Fig. 3(friction factor vs. Mobility Index). Also, the red line with black dots shows the average friction factor vs. Sediment diameter in millimeters for Savkheda gauging station for the analysis (Fig.3).

A. Model IV. Multiple Regression Analysis

Figure 1. Tapi Basin

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TABLE I. MULTIPLYING CONSTANTS FOR MULTIPLE REGRESSION ANALYSIS OF FRICTION FACTOR IN (11)

C1 Q

C2 A

C3 S

C4BW

C5 V

C6 R

C7davg

C8u*

C9 w

0.0306

-1.68E-06

4.73E-05

94.33

-8.83E-05

0.01212

-0.00672

-0.00618

-0.182

0.17291

0.99298

Station Name Savkheda

TABLE II. MATHEMATICAL MODELS USING MOBILITY INDEX FOR TAPI SAVKHEDA MONSOON SEASON

Name of Model Model- I Keulegan‟s Equation

Equation f1 = 0.02867* (MI) 0.23956

Chi2 0.00016

R2 0.94204

Function Allometric1

Weighting method Statistical

Model-II Modified Manning‟s equation

f2= 0.0411* (MI) 0.77775

0.00114

0.96152

Allometric1

Statistical

Model-III Based on multiple regression using factors effecting sediment transportation

f3 = 0.0312* (MI) 0.51019

0.00046

0.87645

Allometric1

Statistical

favg = 0.0325* (MI) 0.37474

0.0003

0.94076

Allometric1

Statistical

0.00037

0.9776

Allometric1

Statistical

Average of f1 to f3

Model-IV Multiple regression Model

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f MR =0.02978* (MI)

0.3197

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Figure 2. Savkheda, Sarangkheda gauging stations and Ukai dam

TABLE III. VARIATIONS OF FRICTION FACTORS O BTAINED BY ALL THE METHODS AND MOBILITY INDEX Friction Factors Mobility Index 0.265 to 0.59

0.0208 to 0.0252

00.0146 to 0.027

0.0158 to 0.0238

fMR

0.0194 to 0.0251

determination (R2) values. The function and the curve which gives near to one value of Co-efficient of determination (R2) are considered as best curve. Also, the value of Chi2 approximately equals to zero gives the best fitted curve. The variations of the friction factor with Mobility Index for Tapi River obtained by all the methods are giver in Table III.

The value of R2(co-efficient of determination) for all the four models shows that Model IV is best for the prediction of the friction factor for Tapi river with co-efficient shown in Table I and the value of R2 for Model IV is 0.99298.

favg

0.017 to 0.0255

V. DATA ANALYSIS In this paper 15 years field data of Savkheda gauging station of river Tapi River is analyzed.

Step: 1 The daily discharge data is converted in to monthly data.

Step: 3 The seasonal data is converted in to yearly data.

Step: 4 The value of friction factor obtained using three approaches. Step: 5 After carrying out multiple regressions on this data results are obtained for monsoon season.

Allometric

0.036

0.034

Allometric

0.032

Allometric

0.030

FRICTION FACTOR

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Step: 2 The monthly data is then converted to seasonal data by taking average of monthly data so obtained is converted in to seasonal data. i.e. monsoon, post monsoon and pre monsoon seasons.

TAPI SAVKHEDA MONSOON

0.028 0.026 0.024 0.022

favg

0.020

Allometric

0.018

fMR

0.016

Allometric

0.014

Allometric

0.012 0.15

Diamter

0.20

0.5

0.25

1.0

Diameter

0.30

0.35

1.5

0.40

2.0

0.45

2.5

0.50

3.0

0.55

0.60

3.5

Step:6 Origin software is used to develop mathematical model to co-relate Mobility Index and friction factor for each approach, average and multiple regression friction factor

Figure 3. Friction factor Vs Mobility Index For Savkheda Station

VI. RESULT ANALYSIS The results of the mathematical model which has been developed relating the Mobility Index with the friction factor are shown in Table II for the friction factor obtained by each method. The Performance of each model (as shown in Table II) is in terms of goodness of fit Chi2 and Co-efficient of

As shown in Fig. 3, when the value of mobility index varies from 0.181 to 0.527, the diameter varies from 1.16 to 3.52 mm, the friction factor f1 obtained by Keuleganâ&#x20AC;&#x;s equation is highest and friction factor f3 obtained by Based on multiple regression using factors effecting sediment transportation is lowest. The pattern of variation for f1 to f3 and favg and fMR

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MOBILITY INDEX

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The variation of friction factor with Mobility Index for averaged and multiple regression friction factor curves appear similar. The friction factor f2 obtained by Modified Manning‟s method give higher value in comparison to the f3 because in computation of f2 the effect of Q, A, n, R, S are considered but the parameters like u/u*, τ*/τ*c, u*/w, Fr, S and R/davg are not considered. When Mobility Index varies from 0.265 to 0.369, the f1 gives highest value of friction factor and the f3 gives lowest value of friction factor. The favg and fMR are higher than f2. f2 is higher than f3.

Mobility Index increases with increase in diameter and friction factor during monsoon indicating increased sediment transportation due to increased flow. Friction factor show almost nonlinear relationship with Mobility Index (Fig.3). When the particle diameter increases the Mobility Index (MI) increases and the friction factor also increases. So, when Mobility Index increases the friction factor increases. By comparing the curve of friction factor by multiple regressions with all other curves at Savkheda station, it can be seen that the multiple regression model gives the maximum value of R2 as 0.9789 and minimum value of Chi2 as 0.00005 using Mobility Index, giving best fit curve. Therefore, the curves obtained by multiple Regression analysis can be used successfully for predicting friction factor of Tapi River at Savkheda as given here. The equations obtained for the prediction of friction factor give relation as

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are similar. The Non linear curve fitter of Origin 7.5 has been used for establishing relation between mobility index and friction factor using Allometric1 function. When the diameter increases from 1.16 to 3.52 mm, the friction factor obtained by all the methods increases ( Fig.3) and Mobility Index also increases from 0.181 to 0.527 (Table III). Almost for all the methods, the variation of friction factor calculated at particular station is almost uniform. An attempt is made under this study to develop simple equations which are best suited for the river under consideration the values used on the basis of average of three different methods can be used without much loss of accuracy. The variation of hydraulic parameters is such that friction factor shows very large variation during monsoon. Looking to such distribution, the best-fit curve based on least square method does not give the satisfactory results or very good value of co-efficient of determination on yearly basis. Therefore, an attempt is made to study seasonal variation of friction factor. Friction factor takes into account the effect of nonuniformity of flow. This nonuniformity can be correlated with bed conditions, flow conditions, dynamic conditions etc. and the relation between friction factor and various non dimensional parameters can be established for given conditions of specific weight of sediment, diameter and sediment characteristics. However, it is very difficult to correlate all the parameters affecting the friction factor. But depending upon local conditions the correlation can be made between friction factor and the variables affecting the sediment transportation. Table II represents the developed mathematical models. The statistical analysis of curve plotted between friction factor and Mobility Index is done by using non-linear-square-fitter to obtain the best fit curve. From the figure 3 depicting Mobility Index vs. friction factor, it is observed that the pattern of variation is same for all methods. Comparison of all the three methods shows that f1 obtained by Kelegun‟s equation and f3 obtained by multiple regreesions using sediment transportation variables deviate maximum. Comparison of all the three methods with corrected averaged friction factor curve and multiple regression curves shows that friction factor by f2 and f3 gives more deviation. An extremely large deviation is observed in case of friction factor f3 curve. VII. DISCUSSION AND CONCLUSIONS

Following findings can be summarized from above study.

A mathematical model is presented to estimate the fraction coefficient in open channel flows. The application of mathematical model may be generalized to in smooth channel as well as in rough channels with free surface flow. The value of R2(co-efficient of determination) for all the four models shows that Model IV is best for the prediction of the friction factor for Tapi river with co-efficient shown in Table I and the value of R2 for Model IV is 0.9776, which is much higher than other three methods. So, (11) can be used successfully for the prediction of resistance to flow for Tapi River. The pattern of variation of Mobility Index with friction factor is same for all methods.

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(12) Mobility Index varies from 0.265 to 0.369 when the diameter increases from 1.16 to 3.52 mm. f MR =0.02978* (MI) 0.3197

Present work also tries in formulating an equation for resistance in alluvial channel comprising all the major parameters affecting the resistance. Mathematically, friction factor can be expressed as a function of Mobility Index. The mathematical model was successfully used to predict the complex non-linear relationship between the friction factor of an open channel flow and its influencing factors. In case of open channel with rigid and movable boundary, Mobility Index affects the flow resistance considerably. So, in any formulation of friction factor, Mobility Index can be included for prediction. Present work shows a way for inclusion of Mobility Index in the analysis and prediction of friction factor. Present work tries in showing a way for formulation of friction factor equation in movable channel. ACKNOWLEDGMENT

It is to be acknowledging that without the permission given by Chief Engineer, Central Water Commission C.W.C., Narmada and Tapi River Basin Organization Baroda; this paper would not have seen the light of the day. The enormous assistance provided by the office of the Executive Engineer Central Water Commission (Tapi Division, C.W.C., Surat) during the preparation of this paper is duly acknowledged.

REFERENCES

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A Wetted cross section area Bw Width of the channel davg Average diameter of sediment in m

ABBREVIATIONS

f1 Friction factor obtained using Keulegan‟s eq. f2 Friction factor obtained by Mul. Reg. by considering the variables like A, R, S, n etc. f3 Friction factor obtained by multiple regression analysis using sediment transport variables favg Friction factor obtained by Average of f1 to f3 fMR Friction factor obtained by multiple regression analysis using all the basic data of Tapi River Fr Froude number MI Mobility Index  's Submerged specific weight of particle h depth of the flow ln(h/davg) Log of Relative flow depth R Hydraulic Mean depth Ss Specific gravity of submerge grain Re Reynolds number υ Kinematic viscosity of water (m2/sec) v Mean velocity of the cross-section w Fall velocity of the particle u* Shear velocity τ*c Critical shear stress τ* Shield Entrainment function  Relative roughness

H. Afzalimehr, “Friction factor for gravel-bed channel with high boulder concentration,” J. Hydraul. Eng., 126(11), 2000, pp. 856- 858. [2] H. Afzalimehr, and F. Anctil, “Estimation of gravel-bed river ﬂow resistance,” J. Hydraul. Engg., 124(10),1998, pp.1054 -1058. [3] A. Bilgil, and H..Altun, “Investigation of flow resistance in smooth open channels using artificial neural networks,”Flow Measurement and Instrumentation, 19, 2008, pp.404-408. [4] V.T.Chow, “Open channel Hydrulics,” Mc Graw Hill Book Co., 1959. [5] C. Colosimo, V. A..Copertino, and M.Veltri, “Average velocity in gravel-bed rivers.” Proc., 5th Congress Asian and Paciﬁc Division of IAHR , 1986,Seoul, Korea, 2. [6] R.J. Garde, and K.G. Ranga Raju, “Mechanics of sediment transport and alluvial stream problems,” Third Edition, New Age International Publisher, 2000, New Delhi, India. [7] G.H. Keulegan, “laws of Terbulent flow in open channels,” U.S.Dept. of Commerce, NBS, vol. 21, Dec 1938,. [8] B. Kumar, and A.R. Rao, “Metamodeling approach to predict friction factor of alluvial channel,” Computers and Electronics in Agriculture 70, 2010, pp. 144–150. [9] M. P. Lamb, W. E. Dietrich, and J. Venditti, “Is the critical Shields stress for incipient sediment dependent on channel-bed slope,”Proc., At JGR-Earth Surfaces, 2008, pp.1- 40. [10] A. Wilcox, J. M. Nelson, and E. E. Wohl, “Flow resistance dynamics in Step-pool channels. 2: Partitioning between grain, spill and woody debris resistance,” Water Resour. Res., 42, 2006, W05419.

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