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Chapter 10: Problem 29

Supercritical, uniform flow of water occurs in a 5.0 -m-wide. rectangular, horizontal channel. The flow has a depth of \(1.5 \mathrm{m}\) and a flow rate of \(45.0 \mathrm{m}^{3} / \mathrm{s}\). The water flow encounters a 0.25 -m rise in the channel bottcm. Find the normal depth after the rise in the channel bottom. Is the flow after the rise subcritical, critical, or supercritical? Assume frictionless flow.

Short Answer

Expert verified
The normal depth after the rise in the channel bottom is represented by \(y_{2}\) which can be found using the formula given. Upon calculating the new Froude number after the rise, it can be identified whether the flow is subcritical, critical or supercritical.

Step by step solution

01

Calculating initial Froude number

Firstly, the Froude number before the rise must be calculated. The Froude number \( F \) in open channel flow is given by \( F = V / (g \cdot y)^{0.5} \) where V is the velocity, g is acceleration due to gravity and y is the depth of flow. The velocity \( V \) is given by the flow rate / (depth * width) = 45/(1.5*5) m/s.
02

Determining the flow

The Froude number gives an indicator of the type of flow. If F < 1 it’s subcritical flow, F =1 it’s critical flow and if F > 1 it’s supercritical. Calculate the Froude number using the values obtained from previous step and determine the type of flow. From this, it can be concluded that the flow before the rise is supercritical.
03

Calculating the sequent depth (y2)

The sequent depth or normal depth after the rise in the channel bottom can be represented as \( y2 = [(y1 \cdot F^{2} + 1) / 2]^{0.5} \). Use this formula to find the normal depth.
04

Identifying the flow after the rise

The type of flow after the rise in the channel bottom can be determined by calculating the new Froude number. The velocity after the rise can be calculated using the same formula as in step 1 but with the updated depth. Subsequently, use this to calculate the Froude number and determine the type of flow.

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Most popular questions from this chapter

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.

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.

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Consider the flow down a prismatic channel having a rectangular cross section of width \(b\). The channel bottom makes an angle \(\theta\) with the horizontal. Show that \\[ \frac{d y}{d x}=\frac{\tan \theta-\left(n^{2} Q^{2}\right) /\left(A^{2} R_{h}^{4 / 3} \kappa^{2}\right)}{1-Q^{2} /\left(A^{2} g y \cos ^{2} \theta\right)} \\] where \(y\) is the vertical fluid depth, \(A=b y \cos \theta,\) ard \(Q\) is the volume flow rate. Discuss the form of the equation for small values of \(\theta\left(\theta \sim 1^{\circ}\right)\)

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