Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 10: Problem 45

Water flows down a wide rectangular channel having Manning's \(n=0.015\) and bottom slope \(=0.0015 .\) Find the rate of discharge and normal depth for critical flow conditions.

Short Answer

Expert verified
The Discharge (Q) and Normal Depth (y_c) under critical flow conditions can be calculated using the given values and the above mentioned steps. The detailed calculation would require substituting the given values into the equations.

Step by step solution

01

Understand the concept of critical flow

The critical flow condition in an open channel is the state at which the specific energy is at a minimum for a given discharge; the depth at this condition is called the critical depth.
02

Using the hydraulic radius in critical flow conditions

In critical flow conditions, for a wide rectangular channel, Critical Depth \( y_c \) equals the Hydraulic Radius \( R \). Hence, the formula for the Hydraulic Radius for a wide channel, which should be equal to \( y_c \), is \( R = \frac{{Area}}{{Perimeter}} = \frac{{by_c}}{{b+2y_c}} = y_c \) where \( b \) is channel width, which can deem to be 1 (unit-width) as it's a wide channel and wouldn't affect our discharge calculation.
03

Using the Froude Number

In critical conditions, the Froude number \( Fr \) is equal to 1. \( Fr = 1 = \frac{V}{\sqrt{gR}} \) where \( V \) is the flow velocity, \( g \) is the acceleration due to gravity and \( R \) is the hydraulic radius. Simplifying for velocity \( V \), we get \( V = \sqrt{gR} = \sqrt{g*y_c} \).
04

Using the Manning's equation

Manning's equation is used to predict the time of flow in a gravity-fed conduit like a ditch or a pipe. It gives the relationship between velocity and hydraulic radius: \( V = (1/n)R^{2/3}S^{1/2} \) where \( S \) is the slope, \( n \) is the Manning's roughness coefficient, and \( V \) is the average velocity. However, in this case, equating the expressions for \( V \) from steps 3 and 4, we get \( \sqrt{g*y_c} = (1/n)y_c^{2/3}S^{1/2} = y_c^{2/3}*S^{1/2}*{1/n} \). Now, we can solve for \( y_c \) by inputting given values.
05

Calculate the discharge \( Q \)

After obtaining the value of \( y_c \), we can calculate the discharge \( Q \) which equals the product of the Area of cross section and velocity, \( Q = A*V = by_c*V = y_c*V = y_c* \sqrt{g*y_c} \)

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Find the discharge per unit width for a wide channel having a bottom slope of \(0.00015 .\) The normal depth is \(0.003 \mathrm{m}\). Assume laminar flow and justify the assumption. The fluid is \(20^{\circ} \mathrm{C}\) water.

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

(See The Wide World of Fluics article titled "Grand Canyon Rapids Building." Section \(10.6 .1 .\) ) During the flood of \(1983,\) a large hydraulic jump formed at "Crystal Rapid" on the Colorado River. People rafting the river at that time report "entering the rapid at almost 30 mph, hitting a 20 -ft-tall wall of water, and exiting at about 10 mph." Is this information (i.e., upstream and downstream velocities and change in depth) consistent with the principles of a hydraulic jump? Show calculations to support your answer.

Water flows in a horizontal rectangular channel with a flowrate per unit width of \(q=10 \mathrm{ft}^{2} / \mathrm{s}\) and a depth of \(1.0 \mathrm{ft}\) at the downstream section \((2) .\) The head loss between section upstream and section (2) is 0.2 ft. Plot the specific energy diagram for this flow and locate states (1) and (2) on this diagram.

An 8 -ft-diameter concrete drainage pipe that flows half full is to be replaced by a concrete-lined V-shaped open channel having an interior angle of \(90^{\circ} .\) Determine the depth of fluid that will exist in the \(V\) -shaped channel if it is laid on the same slope and carries the same discharge as the drainage pipe.
See all solutions

Recommended explanations on Physics Textbooks

Waves Physics

Read Explanation

Magnetism

Read Explanation

Turning Points in Physics

Read Explanation

Work Energy and Power

Read Explanation

Quantum Physics

Read Explanation

Astrophysics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free