Explicit Solution for Critical Depth in Closed Conduits Flowing Partly Full

Haixin Shang,Kuandi Zhang,Song Xu,Luyou Zhao

doi:10.3390/w11102124

Abstract

Critical depth is an essential parameter for the design, operation, and maintenance of conduits. Circular, arched, and egg-shaped sections are often used in non-pressure conduits in hydraulic engineering, irrigation, and sewerage works. However, equations governing the critical depth in various sections are complicated implicit transcendental equations. The function model is established for the geometric features of multiple sections using the mathematical transform method and while considering non-dimensional parameters. Then, revised PSO algorithms are implemented in MATLAB, and the right solution’s formula for the critical depths in various non-pressure conduit sections is established through optimization. The error analysis results show that the established formula has broad applicability. The maximum relative errors of the formula for critical depths are less than 0.182%, 0.0629%, and 0.170% in circular, arched, and egg-shaped sections, respectively, which are more accurate than those of existing formulas; the form of the formula proposed in this work is also more compact than that of the existing formulas. The results of this research may be useful in design, operation, and maintenance in conduit engineering.

Highlights

Circular, arched, and egg-shaped sections are often used in non-pressure conduits in hydraulic engineering, agricultural irrigation, and sewerage works
This paper focuses on the study of the mathematical characteristics of critical depth control equations in circular sections, arched crossing, and egg-shaped sections; constructing the fitting function model with the constraint conditions of the endpoints and derivatives of the equation curve, a general calculation form is obtained, and interpartition calculation can be avoided
This paper presents three explicit solutions for critical depth in closed conduits flowing partly full, they are suitable for circular, arched, and egg-shaped sections, respectively

Summary

Introduction

Circular, arched, and egg-shaped sections are often used in non-pressure conduits in hydraulic engineering, agricultural irrigation, and sewerage works. Because conduits with circular and arched cross-sections have excellent hydraulic properties and convenient construction, circular and arched sections are widely used for free-surface water diversion conduits in hydraulic engineering and agricultural irrigation [1]. Because conduits with egg-shaped sections have functional discharge capacity and compression capability, they are often used in sewerage works in municipal engineering. Conduits carrying sediment and locations where the rock is sheet joint, weak, and very carefully laminated are usually constructed in the form of egg-shaped channels [2]. Critical depths play a significant role in the analysis, design, operation, and maintenance of conduits. In the varied flow computations, one is required to determine critical depths

Results

Discussion

Conclusion