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Chapter 10: Problem 56

A rectangular, unfinished concrete channel of 28 -ft width is laid on a slope of \(8 \mathrm{ft} / \mathrm{mi}\). Determine the flow depth and Froude number of the flow if the flowrate is \(400 \mathrm{ft}^{3} / \mathrm{s}\)

Short Answer

Expert verified
The result of the exercise will be the flow depth \( y \) in ft, and the Froude number (Fr)

Step by step solution

01

Determine Channel Cross Section Area

The first step is to find the cross-sectional area of flow in the channel. This area can be found by using the formula for the cross-sectional area of flow in a rectangular channel, which is \( A = yw \) where \( y \) is the unknown flow depth and \( w \) is the width of the channel, which is 28 ft.
02

Calculate Flow Velocity

Next use the flow rate (Q) and the cross-sectional area of the flow (A) to calculate the flow velocity (V). The formula is \( V = Q / A \) where the flow rate \( Q = 400 ft^3/s \) from the problem, and the cross-sectional area \( A = y * 28 ft \). This likely will give a formula with \( y \) as the only unknown.
03

Apply Mannings Equation for Flow Depth

Using Manning's equation we can solve for \( y \), which is \( V = (1/n) * R^(2/3) * S^(1/2) \), here \( n = 0.013 \) (assuming concrete channel), constant \( R \) is the hydraulic radius, which for a rectangular channel is equal to the flow area (A) divided by the wetted perimeter (P), and the slope \( S = 8 ft/mi \). Convert slope to ft/ft by dividing by 5280. Solve for \( y \) which is the only unknown in the equation.
04

Calculate Froude Number

After finding the flow depth, we can calculate the Froude number (Fr). The formula is \( Fr = V / \sqrt{g * y} \), where \( V \) is the Flow Velocity from Step 2, \( y \) is the flow depth from Step 3, and \( g \) is the acceleration due to gravity (\( g = 32.2 ft/s^2 \)).

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Most popular questions from this chapter

\(\mathrm{A}\) trapezoidal channel with a bottom width of \(3.0 \mathrm{m}\) and sides with a slope of 2: 1 (horizontal:vertical) is lined with fine gravel \((n=0.020)\) and is to carry \(10 \mathrm{m}^{3} / \mathrm{s}\). Can this channel be built with a slope of \(S_{0}=0.00010\) if it is necessary to keep the velocity below \(0.75 \mathrm{m} / \mathrm{s}\) to prevent scouring of the bottom? Explain.

The depth downstream of a sluice gate in a rectangular wooden channel of width \(5 \mathrm{m}\) is \(0.60 \mathrm{m}\). If the flowrate is \(18 \mathrm{m}^{3} / \mathrm{s}\) determine the channel slope needed to maintain this depth. Will the depth increase or decrease in the flow direction if the slope is (a) \(0.02 ;\) (b) \(0.01 ?\)

Find the diameter required for reinforced concrete pipe laid at a slope of 0.001 and required to carry a uniform flow of \(19.3 \mathrm{ft}^{3} / \mathrm{sec}\) when the depth is \(75 \%\) of the diameter.

A rectangular channel \(3 \mathrm{m}\) wide carries \(10 \mathrm{m}^{3} / \mathrm{s}\) at a depth of \(2 \mathrm{m} .\) Is the flow subcritical or supercritical? For the same flowrate, what depth will give critical flow?

A 5.0 -m-wide channel has a slope of \(0.004,\) a \(8.0-\mathrm{m}^{3} / \mathrm{s}\) water flow rate, and a water depth \(1.5 \mathrm{m}\) after a hydraulic jump. Find the water depth before the jump.
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