Introduction
Weirs are simple devices used to measure flow in open canals and ducts. Generally, the geometric cross section of the weirs is rectangular, trapezoidal and triangular. The discharge - height relationship in this type of weirs is nonlinear. The geometry of sharp weirs generally follows a mathematical equation. The governing mathematical equation can also be expanded based on the Gamma function and the Abel integral. The first idea to design the body shape of the weir was proportionally and linearly completed by Stout, 1897 and later by Sutro, 1908. Sutro weir (proportional) consists of a combination of a rectangular section and a special curve. The discharge-height relationship in the Sutro weir is linear and proportional. Sutro type weirs are used to measure discharge in various industries, including chemical industries, wastewater transmission and treatment, and water transfer channels in irrigation and water supply. The geometric dimensions of the base rectangle affect the shape of the curve and the flow rate.

Methodology
In this study, by examining the governing equations and using the expansion of gamma and Abel integral functions, the design equations of sharp overflow body were extracted. The amount of flow through this type of overflow is also a function of hydraulic conditions and geometry of the weir shape body. In this regard, the flow rate passing through these types of weir is important. Dimensional analysis of dimensional factors affecting the Cd discharge coefficient was identified and tested in the laboratory by making 12 different samples of Sutro weir with different dimensions. To make Sutro weir, first using Excel software and placement in the extracted equation, the coordinates of the overflow body are extracted and then using AutoCAD software, the body of the weir is drawn and using a laser cutting machine on aluminum plates perforate with a thickness of 2 mm with precision. Suitable for engraving and cutting. For experiments in the hydraulic laboratory of Eghlid Islamic Azad University, a laboratory channel with a length of 6 meters, width and height of 100 and 60 cm, respectively, with a water circulation system was used. The weirs were installed at the end of the canal and a level gauge with an accuracy of 0.10 millimeter was used to measure the depth above the weir where the water level was completely horizontal. To measure the flow, a Megap magnetic flowmeter with an accuracy of 0.001 liter per second installed in the main transmission pipeline and a standard trapezoidal weir calibrated and installed on a reservoir at the end of the system were used. By establishing the flow, flow and water depth on the weir were measured and finally in all conditions, using the governing relations, the experimental coefficient of flow of each of the weir was extracted.

Results and Discussion
This study conducted to determine the discharge coefficient in the Sutro weir. Dimensional analysis of dimensionless factors affecting the Cd discharge coefficient was identified. Flow rate (Q) diagrams were plotted against the parameter [H-1/3 a] for all experiments. In all graphs, a linear relationship with a correlation coefficient of R2 was observed to be appropriate and near one. The results of dimensional analysis of the parameters affecting the discharge coefficient are as follows equation.
C_d=Q/(2√2g Wa^(1/2) )=f_2 (h/W,h/P,h/a,h/((2W+a)),h/(h+P),R_e,W_b,F_r )
The above dimensionless parameters were calculated for all experiments. In order to evaluate the priority of the effect of dimensionless parameters using SPSS-21 software, this was done first. The results showed that the most effect is related to the dimensionless parameter H / w. Of course, the parameters H / (P + H) and Fr are also in the next priority. Therefore, in the next step, regression model analysis was performed between these three parameters and Cd coefficient. Finally, using multivariate regression, the relationship between flow coefficient and dimensionless parameters was extracted. All tests were performed under turbulent flow conditions. Weber number was also calculated, this number is the measure of the effect of surface tension. The effect of this parameter can be ignored in situations where the water depth is more than about 2 to 3 cm. In addition, the Approach velocity and velocity head were calculated. The results showed that the effect of approaching velocity could also be ignored.

Conclusion
In this research, using the Abel integral function and the expansion of the gamma function in the design of the body shape of the weirs were studied. The first idea to design the body of the overflow was proportionally and linearly completed by Stout, 1897 and later by Sutro, 1908. Various studies have been performed by different researchers in this field, each of which provided equations according to the basic constraints of the weirs. This amount can be determined in the laboratory by making the desired weir and performing the test. In this research, a linear proportional type overflow was made and tested. The analysis of the experiments showed that the value of the weir coefficient discharge in these conditions is equal to 0.620 on average.

Keywords
Linear Proportional Weir, Discharge Coefficient, Abel Integral Function, Gamma Function.