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Chapter 10: Problem 73

Water flows at \(150 \mathrm{ft}^{3} / \mathrm{s}\) in a 3 -ft-wide rectangular cleanearth irrigation canal. The canal slope is \(0.275^{\circ} .\) Al one point, the water depth is \(3 \mathrm{ft}\) (a) Accurately compute the water depth at a location 200 fit downstream. (b) Is the flow at the upstream location subcritical or supercritical? At the downstream location? (c) Sketch the canal and the water surface profile.

Short Answer

Expert verified
The exact downstream water depth would be obtained by solving the modified Manning Equation. The type of flow can be classified based on the Froude number and the sectional view of the canal with the water flow can be sketched accordingly based on the given dimensions and calculated parameters.

Step by step solution

01

Compute the downstream water depth

First, let's find the downstream water depth. In this scenario, since it's a rectangular canal and we know the slope, the discharge, and the initial depth, we can use the Manning Equation to compute the depth downstream. The equation is \(Q=nA(Rh^{2/3})(S^{1/2})\) where \(Q\) is the discharge, \(n\) is Manning's coefficient, \(A\) is the cross-sectional area, \(Rh\) is the hydraulic radius, and \(S\) is the channel slope. For clean earth, \(n=0.022\) and \(Rh\) can be considered effectively equal to the depth as the canal is very wide. This gives us a new equation \(Q=n(width*depth)(depth^{2/3})(S^{1/2})\) which can be solved for depth downstream.
02

Identify the type of flow

Next, we identify whether the flow of water is subcritical or supercritical. This can be done using the Froude number, defined as the ratio of the flow velocity to the wave velocity. If \(Fr<1\), the flow is subcritical (waves can travel upstream) while if \(Fr>1\), the flow is supercritical (waves cannot travel upstream). Velocity \(V\) can be calculated as \(Q/A\) and the Froude number \(Fr\) as \(V / (g * h)^{0.5}\), where \(g\) is the acceleration due to gravity.
03

Sketch the canal

In the final step, we would sketch the canal with its given dimensions and slope along with the water flow. The sketch would consist of a 3 ft-wide horizontal rectangle, signifying the canal, with a water level change from 3 ft to the calculated downstream depth. The water flow direction and whether it's subcritical or supercritical would also be indicated on the sketch.

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Most popular questions from this chapter

A round concrete storm sewer pipe used to carry rainfall runoff from a parking lot is designed to be half full when the rainfall rate is a steady 1 in. $/ \mathrm{hr}$. Will this pipe be able to handle the flow from a 2 -in./hr rainfall without water backing up into the parking lot? Support your answer with appropriate calculations.

A rectangular channel \(3.0 \mathrm{m}\) wide has a flow rate of 5.0 \(\mathrm{m}^{3} / \mathrm{s}\) with a normal depth of \(0.50 \mathrm{m} .\) The flow then encounters a dan that rises \(0.25 \mathrm{m}\) above the channel bottom. Will a hydraulic jump occur? Justify your answer.

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 \(400 \mathrm{ft}^{3} / \mathrm{s}\)

\(\mathrm{A}\) 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.

Water flows in a 10 -m-wide open channel with a flowrate of \(5 \mathrm{m}^{3} / \mathrm{s}\). Determine the two possible depths if the specific energy of the flow is \(E=0.6 \mathrm{m}\)
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