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Chapter 10: Problem 91

A flow of \(873 \mathrm{ft}^{3} / \mathrm{s}\) passes under a sluice gate in \(\mathrm{c}\) rectan gular channel having a gradual contraction in width from 80 fito \(52 \mathrm{ft}\) The channel bed has scoured to a level that is 6 in. below the upstream channel bed, owing to the increased velocity ceused by the sluice gate and the contraction. If the upstream flow depth is \(4 \mathrm{ft}\) and the sluice gate opening is \(1.5 \mathrm{ft}\), compute the depth, velocity, and Froude number of the flow at the end of the contraction.

Short Answer

Expert verified
The upstream and downstream areas can be calculated using the provided dimensions, and then the velocity of the flow upstream and downstream can be calculated using the principle of conservation of mass. The Froude number can then be calculated using these calculated values and the given acceleration due to gravity.

Step by step solution

01

Calculation of Upstream and Downstream Areas

The areas of flow for the upstream and downstream sections of the channel can be calculated using the formula for the area of a rectangle (A = width*depth). The upstream width is given as \(80 \,ft\) and the downstream width is given as \(52 \,ft\). The upstream flow depth is given as \(4 \,ft\), and the downstream flow depth can be calculated by subtracting the scour depth (6 in/12 = 0.5 ft) from the sluice gate opening (\(1.5 \,ft\)). Therefore, the upstream area (A1) and downstream area (A2) can be calculated.
02

Calculation of Velocity

The velocity upstream (V1) can be calculated by dividing the provided flow rate by the upstream area from Step 1. Using the principle of conservation of mass, the velocity downstream (V2) can then be calculated by dividing the flow rate by the downstream area from Step 1.
03

Calculation of Froude number

The Froude number (Fr) can be calculated using the formula: Fr = \(\frac{V}{\sqrt{gD}}\), where V is the velocity of the fluid, g is the acceleration due to gravity (\(g = 32.174 \,ft/s^2\)) and D is the hydraulic diameter (in this case, depth of the flow). The Froude number can be calculated for the flow at the end of the contraction using the downstream velocity and depth from Steps 1 and 2

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