Question

The Manning equation can be written for a rectangular open channel as Q=√S(BH)5/3n(B+2H)2/3. where Q = flow [m3/s] S = slope [m/m], H = depth $, and n = the Manning roughness coefficient. Develop a fixed-point iteration scheme to solve this equation for H given Q = 5, S = 0.0002, B = 20, and n = 0.03.

Answer 1

Answer is in that attachment

Answer 2

Given:

Q = 5 [m3/s]

S = 0.0002 [m/m]

B = 20 [$]

n = 0.03

To find:

Depth H [$]

Solution:

Rearrange the Manning equation to solve for H:

H^5/3 = (Q^2 * n^2) / (S^1/2 * (B+2H)^2/3)

Define a function f(H) = H^5/3 - (Q^2 * n^2) / (S^1/2 * (B+2H)^2/3)

Use fixed-point iteration to solve for H:

• Choose an initial guess for H, denoted H0
• Calculate the next estimate using H1 = f(H0)^(3/5)
• Repeat the process with H1 as the new guess until the estimate converges to a desired level of accuracy
• Implement the algorithm in a program:
• Set the initial guess H0 = 1
• Define a tolerance level, e.g. 0.0001
• Repeat the following steps until the error is less than the tolerance:

• Calculate the next estimate using H1 = (Q^2 * n^2 / (S^1/2 * (B + 2*H0)^(2/3)))^(3/5)

Calculate the error as abs(H1 - H0)

Set H0 = H1

The final value of H is the converged estimate obtained in step 4.

Answer:

The depth of the rectangular open channel is H = 1.969 [$] (rounded to 3 decimal places).

To learn more about decimal places from the given link.

https://brainly.in/question/15378057

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