Critical Depth Estimation in a Channel With Trapezoidal Cross-Section: Proposed Equations, Evaluation of Accuracy and Performance

Estimating the critical depth in a trapezoidal channel, which is widely used in various water conservation and distribution networks, has always been of interest to hydraulic engineering researchers. The common method for calculating the critical depth of a trapezoidal channel is the trial and error method, but today, explicit solutions have been proposed as a powerful and suitable alternative method. In this study, using the dimensional analysis method, different forms of mathematical series and also the two triangle method, equations for estimating the critical depth of a channel with trapezoidal cross-section were proposed and their accuracy was evaluated in comparison with the equations of previous researchers. In extracting the proposed equations, the simple form with fewer repetitions of the mathematical series was considered. A wide range of geometric and hydraulic data of trapezoidal cross-sections at critical depth conditions were used. The accuracy and performance of the proposed and existing equations were evaluated using the absolute of relative error (ARE) and the efficiency index (NSE). The results showed that equations A, B and E with mean absolute of relative error equal to 0.314, 0.173 and 0.078 percent respectively have the best performance in estimating the critical depth of a channel with a trapezoidal cross-section. Also, the accuracy of the proposed equations was evaluated in comparison with the existing equations and the results were reported.
Introduction
One of the most important hydraulic parameters in the analysis of open channel flows, which used in determining the flow pattern, calculating specific energy, analyzing flow stability, designing water transfer structures, and managing flow regime transition conditions is the critical depth (Akan & Iyer, 2021). In most irrigation and drainage network projects, trapezoidal canals are widely used due to their feasibility, good stability, and hydraulic efficiency (Das, 2007). The critical depth equation in trapezoidal canals is a nonlinear and complex equation due to the cross-sectional geometry, which cannot be solved explicitly and requires the solution of related nonlinear equations. In recent years, the development of explicit solutions for determining critical depth has been proposed as a powerful method and a suitable alternative to numerical methods. These solutions are usually extracted using analytical methods, mathematical approximations, nonlinear regression, dimensional relations, or meta-heuristic algorithms, and research results show that explicit solutions, in addition to high speed, have acceptable accuracy and can provide engineers' goals in practical designs and rapid analyses and become a suitable alternative to conventional numerical methods. In the present study, by examining different forms of mathematical equations and also using the dimensional analysis technique and the two triangle method (Arvanaghi et al., 2015), new equations for determining the critical depth of trapezoidal channels were presented in a simpler form and with high accuracy, and the results were compared and evaluated with existing relations.
Materials and Methods
In trapezoidal channels, the critical Froude number equation to calculate the critical depth is (αQ^2 T_c)/(gA_c^3 )=1 (French, 1987). By substituting the A and T relations and changing the variable λ=(myc)/b in the critical depth equation, and by changing the variable φ=〖Qm〗^1.5/(√(□(g⁄α)) b^2.5 ) and substituting it into the critical Froude number equation, the equations φ and λ are obtained. The equation λ, which is 6th degree equation, cannot be solved analytically, but with the help of equation φ and generating φ values for different λ in the applicable range, new equations can be presented to calculate the value of λ and consequently the critical depth. With using the related software and the generated values of φ for λ, and considering the applicable range of Side slope (m) and Botom width (b) values, it is possible to examine different forms of equations such as linear, power, logarithmic, etc. and obtain the relevant coefficients of the equations with appropriate accuracy. To determining the accuracy of equations, two parameters including the efficiency index (NSE) and the absolute of the relative error (ARE).
Results and Discussion
Examining the form of equation 13 (combination of parameter φ with specific powers based on dimensional analysis) showed that equation A had the highest accuracy (with an average absolute value of relative error of 0.314 percent and an efficiency index of 0.995) with 3 iterations of parameter φ and specific powers. In this equation, the maximum observed error was calculated to be 1.738 percent. Next, the power series (equation 14 as a combination of the parameter φ with the power and unknown coefficients) was examined, and in the case of 3 iterations of the parameter φ, equation B with an average absolute value of the relative error of 0.173 percent, an efficiency index of 0.997, and a maximum relative error of 1.611 percent was introduced as the best equation. Examining the different iterations of the series of equation 15 showed that in the case of two iterations of the series, equation D was obtained with an average absolute value of the relative error equal to 0.294 percent, an efficiency index equal to 0.998, and a maximum relative error equal to 2.489 percent. In the following, the three-repetition case of the series was also examined, and the equation E was extracted with an average absolute value of the relative error equal to 0.078 percent, an efficiency index equal to 0.999, and a maximum relative error equal to 0.988 percent, which indicates a significant reduction in the error in the equation E and was introduced as the best relation of this series. Comparing the accuracy of the presented series of equations A to E shows that equation E with the lowest average absolute value of relative error and maximum relative error and the highest value of efficiency index is the best equation for estimating the critical depth of the trapezoidal channel. In the results of the two triangle method with the general form of equation 20, it was observed that the semi-logarithmic equation F with the simple form and the average absolute value of the relative error, efficiency index, and maximum relative error equal to 0.662 percent, 0.985 percent, and 0.922 percent, respectively, has a suitable accuracy in estimating the critical depth of the trapezoidal channel. Comparing the performance of the equations presented by previous researchers shows that the Vatankhah (2013) equation has the highest accuracy among the relations presented and studied, with an average absolute value of the relative error of 0.010 percent, an efficiency index of 0.999, and a maximum relative error of 0.029 percent. Wang's (1998) equation in the form of three series repetitions with an average absolute magnitude of relative error equal to 0.032 percent, an efficiency index equal to 0.999, and a maximum relative error equal to 0.118 percent ranks second in performance next to Vatankhah's (2013) equation. Chen's (2025) equation is also in third rank with an average absolute magnitude of relative error equal to 0.106 percent, an efficiency index equal to 0.998, and a maximum relative error equal to 0.180 percent.
Conclusion
In this study, to calculate the critical depth in a trapezoidal channel, various equations were proposed using dimensional analysis, mathematical series, and the two triangle method, and their accuracy and performance were evaluated in comparison with existing equations of researchers. In extracting new equations and evaluating existing equations, an attempt was made to use equations with minimal repetition of the mathematical series, and the results were compared with the help of evaluation indices including the average absolute value of the relative error, the efficiency index, and the maximum relative error. Among the proposed equations in this study, equations A, B, and E with the average absolute value of relative error equal to 0.314, 0.173, and 0.078 percent, respectively, and also the efficiency index value equal to 0.995, 0.997, and 0.999, respectively, showed good performance in estimating the critical depth of the trapezoidal channel. Among the previous researchers' equations, Vatankhah (2013) equation with an average absolute magnitude of relative error of 0.010 percent and an efficiency index of 0.999 had the best performance in estimating critical depth, and Wang (1998) and Chen (1998) equations with an average absolute magnitude of relative error of 0.032 and 0.106, respectively, ranked second and third. The E equation, proposed in this study with the form of three power series iterations, also had a competitive performance compared to the equations of previous researchers, especially Chen's (2025) equation, with an average absolute value of the relative error of 0.078 percent. The results of this research show that examining different forms of mathematical power series can be of great help in extracting critical depth estimation equations with a simple form and appropriate accuracy.