Question

. Using the Newton Raphson method, determine the uniform flow depth in a trapezoidal channel with a bottom width of 3.0 m and side slope of 1:2 (v:h) with a discharge of 10 (m3/s). The slope is 0.0004 and Manning's roughness is 0.015.

Answer 1

y ≈ 2.5

Step-by-step explanation:

Given data:

bottom width is 3 m

side slope is 1:2

discharge is 10 m^3/s

slope is 0.004

manning roughness coefficient is 0.015

manning equation is written as

`v =1/n R^(2/3) s^(1/2)`

where R is hydraulic radius

S = bed slope

`Q = Av =A 1/n R^(2/3) s^(1/2)`

`A = 1/2 * (B+B+4y) * y =(B+2y) y`

`R =(A)/(P)`

P is perimeter
`=  (B+2√(5) y)`

`R =((3+2y) y)/((3+2√(5) y))`

`Q = (2+2y) y) * 1/0.015 [((3+2y) y)/((3+2√(5) y))]^(2/3) 0.004^(1/2)`

solving for y
`100 =(2+2y) y) * (1/0.015) [((3+2y) y)/((3+2√(5) y))]^(2/3) * 0.004^(1/2)`

solving for y value by using iteration method ,we get

y ≈ 2.5