Estimation of the Velocity Distribution in Circular Pipe with Velocity Measurement at Two Points Using the Renyi Entropy

آب و توسعه پایدار

Article 8, Volume 8, Issue 4 - Serial Number 22, March 2022, Pages 59-68   

Document Type: Original Article/Regular article

DOI: 10.22067/jwsd.v8i4.2107.1060

Authors

maryam teymouri yeganeh^{* 1}; Mohammad mehdi Heidari²

¹Razi UniversityPh.D. Candidate in Water Structures, Water Science and Engineering Department, Razi University, Kermanshah, Iran

²Assistant Professor, Water Science and Engineering Department, Razi University, Kermanshah, Iran

Abstract

In most practical hydraulic engineering problems, accurate flow measurements are required. Understanding flow quantities is an important point in water resources management. Therefore, providing the most appropriate velocity distribution estimation relation that is consistent with the measurement data has always been of interest to researchers. With the development of entropy theory, these methods have been used in a wide range of engineering sciences, including hydraulics and fluid mechanics. In the present study, using the Renyi entropy method, the effective parameter "m" on the Renyi entropy parameter "G" was investigated and the velocity distribution in a circular pipe in the conditions that 36.2, 50, and 70% of the circular pipe fills. It was estimated at two points by measuring the velocity at depths (0.1D-0.9D), (0.2D-0.8D), and (0.3D-0.7D) relative to the water level. In order to determine the accuracy of estimating the velocity distribution using the Renyi method, the correlation coefficient and the root mean square error was used and also to determine the accuracy of entropy parameters, normalized root mean square error was used. The results showed that the Renyi entropy method has high accuracy with observational data. Also, velocity measurement at depth (0.9D-0.1D) from the water surface for 36.2, 50, and 70% of the circular pipe, with the normalized root mean square error to 0.2325, 2.36, and 0.51 respectively, has higher accuracy.

Keywords

Entropy Function Maximization; Velocity Estimation; Lagrange Coefficients; Two-Point Method

References

تیموری یگانه، م. قبادیان، ر. و حیدری، م.م. 1400. مقایسه روابط مختلف برآورد نیمرخ عمقی سرعت طولی بر مبنای روش بهینه سازی با استفاده از الگوریتم ژنتیک. نشریه علمی پژوهشی مهندسی آبیاری و آب ایران، 12(45): 122-137.

Chiu C.L. 1988. Entropy and 2-D velocity distribution in open channels. Journal of Hydraulic Engineering, 114(7): 738–56.
Chiu CL. 1989. Velocity distribution in open channel flow. Journal of Hydraulic Engineering, 115(5): 576–94.
Clark S.P. and Kehler N. 2011. Turbulent flow characteristics in circular corrugated culverts at mild slopes. Journal of Hydraulic Research, 49(5): 676–84.
Cui HJ. and Singh VP. 2013. Two-Dimensional velocity distribution in open channels using the Tsallis entropy. ournal of Hydraulic Engineering, 18(3): 331–9.
Jiang Y., Li B. and Chen J. 2016. Analysis of the Velocity Distribution in Partially-Filled Circular Pipe Employing the Principle of Maximum Entropy. PloS ONE, 11(3): e0151578.
Kumbhakar M. and Ghoshal K. 2016. Two dimensional velocity distribution in open channels using Renyi entropy. Physica A: Statistical Mechanics and its Applications, 450(15): 546-559
Kumbhakar M. and Ghoshal K. 2017. One-Dimensional velocity distribution in open channels using Renyi entropy. Stochastic Environmental Research and Risk Assessment, 31(4): 949–959.
Luo H. and Singh VP. 2011. Entropy theory for two-dimensional velocity distribution. Journal of Hydraulic Engineering, 16(4): 303–15.
Maghrebi M.F. 2006. Application of the single point measurement in discharge estimation. Advances in Water Resources, 29: 1504-1514.
Marini G., Martino GD., Fontana N., Fiorentino M. and Singh VP. 2011. Entropy approach for 2D velocity distribution in-open channel flow. Journal of Hydraulic Research, 49(6): 784–90.
Rahimpour M. and Maghrebi M.F. 2006. Prediction of stage–discharge curves in open-channels using a fixed-point velocity measurement. Journal of Flow Measurement and Instrumentation, 17: 276-281. 
Renyi A. 1961. On measures of entropy and information, in: Proceedings, 4th Berkeley Symposium on Mathematics. Statistics and Probability, 4.1: 547–561.
Shannon CE. 1948. A mathematical theory of communication. Bell System Technical Journal, 27(7): 379–423.
Singh V.P. and Luo H. 2011. Entropy theory for distribution of one-dimensional velocity in open channels. Journal. of Hydrologic Engineering., 16(9): 725-735.
Sterling M. and Knight D.W. 2000. Resistance and boundary shear in circular conduits with flat beds running part full. Proc. ICE Water Maritime Energy,.142(4): 229–40.
Teymouri Yeganeh M. and Heidari MM. 2020. Estimation of one-dimensional velocity distribution by measuring velocity at two points. Flow Measurement and Instrumentation, 73(2020): 101737. 
Yoon J.I., Sung J. and Lee MH. 2012. Velocity profiles and friction coefficients in circular open channels. Journal of Hydraulic Research, 50(4): 304–11.