Flow Speed in a Channel The speed of water flowing in a...

Flow Speed in a Channel The speed of water flowing in a channel, such as a canal or river bed, is governed by the Manning Equation, $$ V=1.486 \frac{A^{2 / 3} S^{1 / 2}}{p^{2 / 3} n} $$ Here $V$ is the velocity of the flow in $f / s ; A$ is the cross sectional area of the channel in square feet; $S$ is the down- ward slope of the channel; $p$ is the wetted perimeter in feet the distance from the top of one bank, down the side of the channel, across the bottom, and up to the top of the other bank); and $n$ is the roughness coefficient (a measure of the roughness of the channel bottom). This equation is used to (a) Find the speed at which water flows through the canal. (b) How many cubic feet of water can the canal discharge per second? IHint: Multiply $V$ by $A$ to get the volume of the flow per second.]

Flow Speed in a Channel The speed of water flowing in a channel, such as a canal or river bed, is governed by the Manning Equation,
$$ V=1.486 \frac{A^{2 / 3} S^{1 / 2}}{p^{2 / 3} n} $$
Here $V$ is the velocity of the flow in $f / s ; A$ is the cross sectional area of the channel in square feet; $S$ is the down- ward slope of the channel; $p$ is the wetted perimeter in feet the distance from the top of one bank, down the side of the channel, across the bottom, and up to the top of the other bank); and $n$ is the roughness coefficient (a measure of the roughness of the channel bottom). This equation is used to
(a) Find the speed at which water flows through the canal.
(b) How many cubic feet of water can the canal discharge
per second? IHint: Multiply $V$ by $A$ to get the volume
of the flow per second.]

Key Concepts

Flow Discharge

Flow discharge is the volumetric flow rate, representing the volume of water passing through a given cross-sectional area per unit time. It is calculated by multiplying the flow velocity by the cross-sectional area, and it is a fundamental concept for assessing the capacity and performance of a channel.

Roughness Coefficient

The roughness coefficient quantifies the frictional resistance of the channel surface against the moving water. This parameter is essential for accurately predicting flow velocities as it accounts for the texture and obstructions found on the channel bottom and banks.

Channel Slope

The channel slope is the measure of the steepness or decline of the channel's bed. It represents the component of gravitational force acting to move the water and is a key factor in determining the velocity and energy of the flow.

Manning Equation

The Manning Equation is a formula used in open channel hydraulics to estimate the average velocity of a flowing fluid, such as water in rivers or canals. It relates the channel geometry and roughness to the gravitational force driving the flow, thereby providing a practical and widely-used method to predict flow behavior in natural and engineered channels.

Cross-sectional Area

The cross-sectional area is the area of the channel’s flow section perpendicular to the direction of the flow. It is a critical parameter because it directly influences the volume of water that can pass through the channel at any given time, affecting calculations of discharge.

Open Channel Flow

Open channel flow refers to the movement of a liquid with a free surface, where the upper boundary is open to the atmosphere. This type of flow is distinct from pipe flow and is governed by gravitational forces, making it key in studying rivers, canals, and other surface water systems.

Wetted Perimeter

The wetted perimeter is the length of the boundary of the channel that is in direct contact with the flowing water. It plays a crucial role in determining the hydraulic radius, which in turn affects the resistance to flow and overall efficiency of the channel.

Transcript

00:01 Hey guys, today we're going through problem number 96 on page 31 of the textbook.

00:07 So this is a word problem and i've written out the given information in black on the top.

00:15 So they give you this formula, v, where v is the velocity, and they tell you that is in feet per second.

00:28 And they give you 1 .486 times in the numerator 8 to the power of 2 over 3 times s to the power of 1 over 2.

00:36 And in the denominator, they give you p to the power of two -thirds times n, which is the same as n to the first power.

00:45 And then after reading through what these variables represent, they give you values for them in the canal.

00:53 So we see that a is equal to 75 feet squared.

00:57 Again, watch your units.

00:59 S is 0 .05.

01:00 They don't give units for that.

01:01 P is the perimeter, so they give you 24 .1 feet, and then n is 0 .04.

01:08 And again, remember the velocity is in feet per second.

01:11 So we're just going to jump right into part a, which asks us to find the speed.

01:16 Now, the first thing you're going to want to notice is that the speed is the same thing as v, speed and velocity, because in order to find the speed, we want that feet per second.

01:27 And that's what you would get with your velocity here.

01:31 So we're going to use that equation, 1 .486.

01:36 And they ask for the canal.

01:39 So the important thing here is to simply show that you're using this formula up there, right here, that i've underlined.

01:50 Show that you're using that formula.

01:51 And again, they ask for the canals.

01:52 You're going to be plugging in all four of these values.

01:57 So in the numerator, so i'll write this in parentheses, one big parentheses.

02:02 In the numerator, you have 8 of the 2 thirds, so 75 to the 2 thirds power.

02:14 Times s to the one half power and s to the one half power is 0 .05 oh, which i just wrote as 0 .05 up there, right here, to the one half power right there.

02:28 And then in your denominator, sorry, that's a two.

02:33 Let me rewrite that.

02:34 It's a little messy.

02:39 So this is to the one half power.

02:41 And then in the denominator, what you're going to have is p to the two -third power times n or n to the first power.

02:46 So p to the two -third power.

02:48 So p to the two -third.

02:48 Power is going to be right here 24 .1 to the two -thirds and then end to the first power is the same thing is just end by itself.

03:02 So i'll just write a 0 .04 there and there you have your equation.

03:09 Now what i would do is i would work with this.

03:13 Use a calculator to figure out what this is equal to.

03:16 So do 75 to the two -thirds first...