圓形渠道水質模式之研究

A Study of Water Quality Model for Open Channel Flow with Circular Cross Section

謝瑋珍(Wei-Chen Hsieh)

指導教授 ： 王士紘

淡江大學/工學院/水資源及環境工程學系碩士班/碩士(2005年)

https://doi.org/10.6846/TKU.2005.00655

摘要

本旨在建立污染物濃度擴散問題之線性解析解法。藉由此解析解可得水質模式中污染物擴散問題，提供工程師設計參考。
聖凡納方程式包括水流連續及動量方程式，配合濃度連續方程式得出水質模式。由於水質模式是非線性微分方程式，無法求得解析解，多採用數值解法，但數值解法只是近似解，有不穩定、發散及誤差的問題。本文提出線性解析解法，可用在求解及推導水流及濃度擴散方程式係數及其模式之解答。
線性解析解法曾成功的使用在求解矩形、寬矩形渠道及天然河道之洪流演算。由於圓形渠道較矩形、寬矩形渠道複雜，過去前人使用泰勒展開級數推導出擴散方程式之係數並求解，然而泰勒級數項數是無限的，因此較難得出精確解。本文提出水質模式線性解析解法，不需使用無限項之泰勒展開級數，只要利用三角函數代入方程式，微分計算後，就能得到擴散方程式係數之詳細推導，所得之係數能使解析解更為精確；在污染物濃度擴散問題上，亦可迅速地求解出較佳的線性解析解。
本文則利用參考流量以試誤法推估參考圓心角，再將所得之參考圓心角代入修正後之係數，並將線性化之係數設為定值，求出涵管渠道流量與污染物質分佈情形。同時亦進行參考流量、參考圓心角、渠底坡度、謝塞係數等線性化因子之敏感度分析。以得到最佳解析解時各參數之適用範圍。所發展出的線性化解析解具有效率、簡易及精確之工程實用價值。

關鍵字

線性解析解 ； 聖凡納方程式 ； 洪流演算 ； 污染擴散

並列摘要

The objective of this study is to develop a linearized analytical model and solve the diffusion problems of pollutants.
The water quality model includes unsteady discharge equations and continuity equation of concentration. Because the De Saint Venant equations which contain the continuity and momentum equations, and the continuity equation of contraction are nonlinear partial differential equations, it is difficult to obtain the exact solution. In this study, a linearized analytical technique is provided, not only apply to the scheme of the nonlinear differential equations, and also apply to the coefficients of the diffusion terms of the water quality model.
The linearized analytical method has been used in solving regular rectangular cross sections of open channel successfully. Because the circular cross section of the open channel is more complicated than other rectangular or wide rectangular cross sections, the methods used before of obtaining coefficients of the diffusion equations of discharges and concentrations are using Taylor Series expansion which exist the infinite terms of the series. Therefore, it is impossible to obtain accurate solutions. In this study, an unsteady water quality model of open channel with circular cross section is proposed. By using the linearized analytical method to modify the coefficients of the diffusion equations of and concentration, the good results are obtained.

並列關鍵字

De Saint Venant equations ； open channel ； circular section ； water quality model

參考文獻

(3)Dooge, J.C.I, Kundzewicz, Z.W. and Napiorkowski, J.J, “On Backwater Effects in Linear Diffusion Flood Routing”Hydrological Science, 28.3.9.1983,pp391-402.

(11)Orlob, G.T. “Mathematical Modeling of Water Quality：Stream, Lakes, and Reservoirs”, 1972.

參 考 文 獻

(1) Hayami, S.,“On the Propagation of Flood Wave”, Kyoto University Disaster Prevention Research Institute, Bulletin No.1, Kyoto, Japan, 1951.

(2)Ponce, V.M. and Simons, D.B, “Shallow Wave Propagation in Open Channel Flow” Journal of the Hydraulics Division, ASCE, Vo1.103, HY12, Proc. Paper13392, Dec, 1977, pp1461-1476.