Exact Solution of Optimum Hydraulic Power-Law Section with General Exponent Parameter

Abstract

Power-law sections provide great flexibility in open channel design. However, in the literature the optimum hydraulic power-law section has been developed for only specific values of the exponent k of the power-law formula. This paper presents a general exact solution of the optimum hydraulic section with k as a parameter based on Gaussian hypergeometric mathematics and the Lagrange multiplier method. The relationships between k and each of the optimum width-depth ratio and the side slope are derived. The explicit exact formulas of the shape factor, normal depth, critical depth, discharge, wetted perimeter, and flow area for different k values are presented. The results show that the discharge of the optimum hydraulic section increases as k≤3.3 and then decreases as k≥3.4 for a given flow area or wetted perimeter. In addition, a super-best hydraulic power-law section with k=3.3471 exists, where the discharge is largest. This super-best section represents a new discovery as it provides the global maximum discharge among all possible power-law section shapes. The characteristics of the super-best section are presented.