Calculating pipe diameter for half-flow condition/mannings equation

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Hello,

My knowledge on fluid mechanics is on a introductory level. When explaining, it would be great if you could explain in simple English and also like teaching to a student who isn't very good at the subject area!
I've been reading a lecture slide and have been trying to follow the example to solve my given problem. I don't really understand how the lecturer got certain parts of the solution though (equation hasn't been explained/defined on any of the slides). I don't understand how the lecturer found the diameter or the suitable velocity is for half-flow.

EDIT: The question doesn't require me to find whether the calculated velocity is suitable, but I would like to know how you do find it.

I have attached the slide that I'm using as an example below.

I used A = pi*D^2/8 and P = pi*D/2 as my problem is for half flow condition.

What is wrong? How do I do this correctly?

Thank you for reading. Any help would be appreciated, thank you.

Homework Statement



If water flows from Point D at Kinglake to Point E at Sugarloaf in a lined circular channel laid on a slope of 0.001. If Manning’s n = 0.013 and if the design discharge is 0.07 m^3/s, what diameter channel should be designed for a half-flow condition? What will be the mean velocity in the channel for this condition?

Manning’s n = 0.013
Slope = 0.001
Q, Discharge = 0.03m^3/s

Find diameter of channel for half flow condition.
Find mean velocity of channel in half flow condition.

Homework Equations



Mannings equation:[/B]

Manning's equation written as Q = (1/n) A R^(2/3) S^(1/2), with Q on the left and n, A, R and S on the right.

Lecture slide with the worked example for a half-full circular pipe under Manning's equation, showing the solution steps and formulas for diameter and velocity.

Handwritten or printed lecture slide equation: Manning's equation with symbols n = n value for channel surface

Plain text formula: A = Flow area (m^2)

Equation R = HydraulicRadius(m), defining the hydraulic radius R in metres as used in Manning's equation.

Text-only equation: S = ChannelSlope, with S on the left and ChannelSlope on the right.


Velocity equation:

Equation for mean velocity: V = Q/A, with V on the left, Q over A on the right.

Handwritten-style lecture slide showing a Manning's equation worked example with the given values and steps for solving for channel diameter and mean velocity under half-flow conditions.

Text equation: V = flow velocity, m/s

Equation: Q = discharge, m^3/s

Text-only snippet showing the area formula A = Area.


The Attempt at a Solution



Manning's equation written as Q = (1/n) A R^(2/3) S^(1/2), where Q is discharge, n is Manning's roughness coefficient, A is cross-sectional area, R is hydraulic radius and S is slope.

Handwritten calculation of AR^(2/3) = Qn/S^(1/2) = (0.07 m^3/s x 0.013)/(0.001^(1/2)) = 0.02877 m^(8/3), using Manning's equation.

abl9oi.gif

A = pi*D^2/8 and P = pi*D/2
Diamter, D = 0.53m^(8/3)

How do I find the diameter required for the channel from this? Is this (above) the diameter? I don't understand on the attachment below, how 0.52m^(8/3) becomes a diameter of 1.21 metres?

------

Formula for flow velocity written as V = Q/A, where V is velocity, Q is discharge and A is cross-sectional area.

V = 0.07m^3/s/((pi*(0.53)^2)/8) = 0.634 m/s

How do I find whether the velocity I calculated is suitable?
 
Attachments
  • Worked solution slide for sizing a circular concrete sewer pipe using Manning's equation, showing Q = 3.1 m3/s, S0 = 0.006, n = 0.013, the result D = 1.21 m rounded to a 125 cm pipe, and a velocity check of 2.53 m/s.
    manningseq.png
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You can't change the formula for the perimeter and/or area of a circle. If you want a half flow condition than divide the flow by 2.
 
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