107 geometry of a trapezoidal channel section b p

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Unformatted text preview: analysis of the trapezoid section will show the basic results. Consider the generalized trapezoid of angle in Fig. 10.7. For a given side angle , the flow area is A y2 by cot (10.22) The wetted perimeter is P b 2W b 2 1/2 ) (10.23) 2 1/2 (10.24) 2y(1 Eliminating b between (10.22) and (10.23) gives P A y y 2y(1 To minimize P, evaluate dP/dy for constant A and A 2 2 1/2 y [2(1 ) P ) and set equal to zero. The result is 2 1/2 4y(1 ) 2y Rh 1 2 y (10.25) The last result is very interesting: For any angle , the most efficient cross section for uniform flow occurs when the hydraulic radius is half the depth. Since a rectangle is a trapezoid with 0, the most efficient rectangular section is such that 2y2 A P 4y Rh 1 2 y b 2y (10.26) To find the correct depth y, these relations must be solved in conjunction with Manning’s flow-rate formula (10.19) for the given discharge Q. Best Trapezoid Angle Equations (10.25) are valid for any value of . What is the best value of for a given depth and area? To answer this question,...
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