General Formulation of Best Hydraulic Channel Section

The best hydraulic channel section is determined by using Lagrange's method of undetermined multipliers. For a given flow, roughness coefficient, and longitudinal slope, this method optimizes the channel section by minimizing the wetted perimeter (or the cross‐sectional area) subject to a constraint. Any flow equation, e.g. Manning, can be used as the constraint. The channel section could contain any number of variables; e.g., two variables (rectangular and triangular sections), three variables (trapezoidal and round‐bottom triangular sections) and so forth. The approach presented is more general than the conventional methods given in the textbooks. It is shown that minimization of the wetted perimeter and minimization of the cross‐sectional area are mathematically equivalent. The method is applied to the standard sections as well as the round‐bottom triangular section. The best hydraulic round‐bottom triangular section, the determination of which is made possible by this approach, is slightly more efficient than the similar and more widely used trapezoidal section. The geometric properties of the best hydraulic round‐bottom triangular section are of great interest. Its depth is equal to the round‐bottom radius and is twice its hydraulic radius. The proposed method can be applied to other complicated sections that cannot be solved by the traditional method.