CE 151 -Spring 2012
Manning's Equation and Newton's method
Assigned Monday, 20 February 2012
Due Wednesday, 29 February 2012, 8:30 AM
Work this assignment in a spreadsheet, and hand in neatly formatted, well-organized printouts of
your spreadsheet work. Be sure that your name is computer-printed on each page that you hand
in. Follow the Given ... Find ... Solution format presented in class.
Recall that the Manning Formula estimates the flow in a channel given the channel's geometry
as indicated below:
k= dimensional coefficient, [L1I31T], 1 m1l31s = 1.486 f1l3/s
z = Side slope, dimensionless, RunlRise
B = Bottom width, [L]
n = Manning roughness, dimensionless
S = Energy (bed) slope, dimensionless, RiselRun
d Flow depth, [L]
A = Cross sectional area, [L 2]
WP = Wetted perimeter, [L]
Q = Flow rate, [L3/T]
A Bd+zd2
WP = B+2d..J);;.2
2
v=~(~)3JS
n WP
Q vA =!5..A5I3Wp-2/3JS
n
1. Given: A channel with a bottom width B of40.0 feet, side slopes z of 1.25, a flow depth d of3.72 feet, a bed slope S of0.0010, and a roughness n of 0.022. On this problem, follow the rules for units and sig figs.
Find:
A. The cross-sectional area of flow A.
B. The wetted perimeter WP.
C. The velocity v.
D. The flow rate Q.
2. Given: The same channel as stated in Problem 1, but with flow rate Q = 600 Pis.
Find: Using Newton's method, determine the flow depth d. Find d so that Q is within 0.001 Pis of 600 f3/s. Show Q and d both rounded to the appropriate number of sig figs and to at least 3 decimal places to demonstrate that you are within 0.001 f3/s ofthe target flow. State your initial
guess and your value ofh for Newton's method as described below. Pick h = 0.01, and start with an initial guess of2 ft.
Recall that the algorithm for Newton's method to find the root of some function f{x) is:
1.
Guess x.
2.
Evaluate f(x).
a.
Iff(x) is close enough to 0, x is the answer, and you're done.
b.
Otherwise you need to improve your estimate of x:
i. Approximate the slope of f(x) as slope = [f{x+h) -f(x)]/h, where h is a small number, say 0.01.
11. Improved estimate ofx = x -f(x)/slope. Iii. Go through step 2 again.
You'll iterate until you are "close enough" to zero with f(x).
Note -If you set up your solution for Problem 2 the right way, you can check your coding ofthe Manning formula by entering 2.72 feet for d, and be sure you get the same answer you got in Problem 1. Also, you can check your final answer for depth in Problem 2 with your calculations in Problem 1 and be sure you get 600 ft3/s.
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