Discharge Coefficient Calculation for Skew Side Weir

Abstract

In irrigation and drainage structures, side weir is widely used for flow diversion from main to branch channels. Side weir is also can use as a measuring device for discharge measurements, so, discharge coefficient was mainly studied in many previous literatures. Skew side weir was not taking a good highlight in previous studies and literature, so the present work discharge coefficient calculation for the skew side weir was adopted and studied. MLR and GEP tools were used in the present study and compared with observed values of Cd.

The mean absolute error for Cd observed and calculating in MLR and GEP was not exceed 5%.

The Cd values for skew side weir ranged from (0.65) to (0.85) while its values for straight vertical side from previous literature weir ranged from (0.45) to (0.65) this mean skew side weir can be used for increasing discharge diversion to the branch channel at the same water levels by 27%. The Akaike information criteria (AIC) with (AICs), root mean square error (RMSE), mean absolute relative error (MARE) and scatter index (SI) are used in this study for measuring the GEP model performance.

Get quality help now

Bella Hamilton

Verified writer

Proficient in: Science

5 (234)

“ Very organized ,I enjoyed and Loved every bit of our professional interaction ”

+84 relevant experts are online

Hire writer

From results the GEP model has AIC=-216.51, AICs=-918.51, RMSE=0.004653, MARE=0.005234, R2=0.994 and SI=0.006231 performed the best. According to previous results the new equation presented through GEP can be adopted for discharge coefficient calculation in skew side weir.

Introduction

Side weirs are an overflow weir installed on the side of the main channel, which allows flow when water rises above the crest. This type of flow considered as a spatially varied flow.

Side weirs usually used as a control structure and as head regulators in irrigation structure. Many literatures deal with side weir hydraulics some of these study deals with sharp crested side weirs such as El-Khashab and Smith (1976), Uyumaz and Smith (1991) Swamee et.al. (1994), Hager (1987), Masoud G. (2003), Singh et.al. (1994), Rao and Pillia (2008), Delkash and Babak (2014) and other investigated deals with inclined and oblique side weir such as Mwafaq and Ahmed (2011). Honar and Javan (2007) and Amir et.al. (2016). The numerical analysis on inclined side weir were investigated by Ahmed (2011), Ahmed et.al. (2013) and Ahmed (2015).

A powerful tool recently used to solve complex nonlinear and multi linear regression equations in hydraulic engineering such as Artificial neural network ANN, genetic programming (GP) (Kisi et.al.2012), Ahmed Y.M. (2018) and Hana et.al. (2019). In recent years used the GEP to model of nonlinear problems. Isa et.al. (2015). The aims of this study are estimates MLR equation and compared with equation modeled from GEP for coefficient of discharge calculation from skew side weir then compared these values with values of Cd estimated from the rectangular side weir.

Experimental Methodology

According to Azza,2012 the experimental works were achieved in the laboratory of hydraulics in Mosul University. The channel was in rectangular cross section at 10m long,0.3m wide and 0.45m depth while the side channel dimensions were 0.15m wide 0.3m depth and 2m long. The discharge was measured using standard sharp crested was (0.15*0.3*0.01) m dimensions at main channel end with sharp crested lip inclined at (450), the side weirs were fixed at the entrance of the side channel by different angles starting from (900) (perpendicular to the side channel) decreases to (300). Five different angles were taken (900,750,600,450 &300) inclined to the (left of flow direction).

Equation can be calculated by trial and error from volumetric calculations. The actual discharge was measured by closed side channel and measured depth of water over the standard weir at the end of main channel, then from equation 1 found Q1, then open side channel and measured water depth over the standard weir at the end of the main channel once again and from equation 1 found the discharge again, but discharge measured in this case (when side channel open) was Q2. To find actual side channel discharge Q3 used equation 2.

Theoretical Methodology

In the general flow through the side weir were derived depends on head of water over the side weir as well as the velocity of flow through it, according to specific energy assumption (De-Marchi 1934).

Depending a De-Marchi, Equation 4 can be written as:

q=discharge per unit length, S=longitudinal slope, =coefficient of discharge

Equation satisfy for rectangular channel and side weir perpendicular to channel bed, so in skew side weir where it’s not perpendicular to channel bed, the angle for inclined side weir must have taken then equation 5 must change depends on these angles.

Dimensional Analysis

De-Marchi equation  was used to calculate Cd from standard side weir in rectangular channel so, dimensional analysis was important to study the effects of the angle skew side weir. The parameters involved is calculated Cd in skew side weir was: Fr= Froude number .

There are many applications for involved regression analysis. These applications deal with linear and nonlinear analysis, depends on variables involves in the problem. The multiple regression model MLR for examples of these equations obtained when more than one variable involves. In order to obtain a general equation for skew side weir, several trials with several equation models examined using (Statistical Package Social Sciences SPSS user guide)

According to (Ahmed 2015) From equation 7 using several models of SPSS, equation 8 can be developed as MLR with a coefficient of determination R2 (0.958)

Programming using Gene expression was an artificial procedure to solve genotype system. This way was invented by Ferreira 2001, 2006, GEP was similar to (GA) genetic algorithms and (GP) genetic programming, in GA deals with individuals as a linear string of length fixed (chromosomes) while GP deals with individuals as nonlinear entities for different parse tree structure. In GEP the individuals deal as encoded linear strings (chromosomes) which are expressed as nonlinear entities. In GEP there are two important players, the Tree structure (ETS) and chromosomes. The decoding of the process information is called translation, then implies obviously a type of code and rules.

The genetic code of GEP was simple, a relation between the symbol of the chromosomes and the node represented in the tree. The rules of GEP determine nodes in the trees, then the type of the interaction in sub-ETS. GEP programming depends on two principal languages, the genetic language and expression trees language. This bilingual notation in GEP is named Karva.

The steps of GEP shown in figure 3, the procedure, including some steps at the begin with the randomly generate of the chromosome from initial population. Then these chromosomes were expressed and exclude the tree expression to evaluate fitness. The individual then selected with respect to their fitness to reproduce with the modification, these individuals are subject to the same development. This process was repeated several times until found a good solution. (Ferreira 2004) the basis of GEP establishes on the structure of GEP gene. The simple structure of genes allows the encoding of thinkable program and allows their dynamic evolution due to these multilateral structural arrangements, a power full set of genetics worker can be implemented to search efficiently solution Ferreira 2002.

Results and Discussion

The genetic operation parameters setting were presented in table 1, while table 2 represents the statistics obtained from GEP after tested more than 1000 equation models and running more than eleven hours. The comparison between the results of the GEP and MLR presented in this study as well as MLR for previous studies illustrated in table 3 are presented in terms of coefficient of determination (R2), Root Main Square Error (RMSE), Akaike Information Criteria (AIC), (AICs – which AIC with a correction for small sample sizes), Mean absolute Relative Error (MARE) and Scatter Index (SI).

Results represents statistics comparison for the present work with previous studies, it may be seen that the GEP model refers to highest value of R2 (0.994) and the lowest value of MARE and RMSE (0.00523 and 0.00465) respectively, as well as the AIC refers to the best value (-216.51) compared with all others equations, all that indicate that the execution of GEP is the best with respect to other previous equations, over all, all values refer to a good agreement of equation for the present work compared with MLR according to Ahmed (2015) and all other previous equations.

Conclusion

In the present study a Gene Expression Programming (GEP) was used to predict an equation for calculating coefficient of discharge in skew side weir in a rectangular channel, this equation was compared with equation predicted from Multiple Linear Regression (MLR) which estimated from statistical tools, the two methods give a good results compared with the observed one with absolute error not exceed 5% for both methods with correlation coefficient 0.9 and 0.996 for MLR and GEP respectively, as well as the Root Mean Squared Errors (RMSE) was 0.0123 and 0.0046 for MLR and GEP respectively.

The results presented in this method compared with others equation calculated show that the accuracy of modeling and fitting of GEP is better than other methods. This conclusion is made by considering the fact that the AIC for the number of parameters that fitted in model which its value (-216.51) is the best value compared with other equations, as well as the best value of present work model for (AICs=-918.51, MARE=0.005234 and SI=0.006231) compared with other values of equations presented in this study. The values of Cd for skew side weir was greater than its values for straight vertical. Finally, the results refer to using GEP gives more accuracy than MLR and other previous literature equations in discharge coefficient calculation and may be used as an improved alternative technique.