General explicit solutions of most economic sections and applications for trapezoidal and parabolic channels

A channel section that has minimum construction cost is known as the most economic section. Such a section has important implications for economic efficiency. However, the most economic section is a complex optimization model with nonlinear objective function and constraints that is difficult to use by ordinary engineers. A general simple formula for the most economic section has not been attempted. In this paper, the general differential equation for the most economic section is derived using Lagrange multiplier optimization method. A simple method to solve the most economic section is proposed that converted the optimization model into a general equation for the most economic section of any shape. By solving this equation, the dimensions of the most economic section are directly obtained. To illustrate, the direct formula for trapezoidal section is derived. To aid application in practice, a simple explicit iterative formula for trapezoidal sections is presented. The direct and explicit iterative formulas were validated. The proposed method is superior to the classical optimization method and as such represent a valuable tool for open channel design. To illustrate the versatility of the presented method, a direct formula for the parabolic section was also derived.