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The Slope Area Method uses the slope of the water and the area of a cross-section to calculate the discharge, or amount of water that moves though a particular point of the waterway. This information is key for hydrologist and hydraulic engineers, who help design and build canals and other bodies of water.
Below is the equation, called the Manning Formula, to calculate the discharge using the Slope Area Method.
Q = (1.49/n)A(Rh^2/3)S^1/2
This equation looks a bit complicated but it's not so bad once we break it down.
Q = the discharge
A = is the area of the cross-section
S = the slope of the waterway
n = Mannings Roughness Coefficient
Rh = the hydraulic radius of the cross-section
Now, let's look at each of these components a bit closer, to help us understand how to use this formula to get the discharge.
A = the area of the cross section
First off, let's talk about what a cross-section is. A cross-section is the surface that would be exposed if one were to make a straight cut though something. Here is an example of a cross-section of a man-made canal and the area of that cross-section must be calculated. For simplicity, we will use the semicircle shape for the examples in this lesson, but other shaped channels are common as well, such as the trapezoidal, rectangular, and triangular.
Area of a semicircle = pi x R² ÷ 2
pi = 3.14
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The next two parts, the slope and the Mannings Coefficient, both illustrate why this method of calculating the discharge of a body of water works best with man-made bodies of water, not natural rivers, streams, channels, lakes, etc. Let me explain what I mean:
In order for the formula to work, certain assumptions must be made about the characteristics of the body of water that are not realistic for natural channels, such as the water flow is assumed to be even throughout and the lining at the bottom of the water is assumed to be uniform throughout. In natural bodies of water, many factors effect the flow of water and the bottom of he channel, such as plant life, fish, natural obstructions, etc.
S = the slope of the water
Those setting out to calculate the discharge of a particular channel of water, must find a uniform reach of the channel, where the water flow is constant in every way, meaning the volume of water flow is constant and free of any obstructions. It would be quite difficult to calculate the slope (rise over run) if these conditions are not met.
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n = the Manning Roughness Co-efficient
This number is a constant based on the material(s) used as the lining or bottom of the waterway, such as concrete, bricks, or lead. Below is a chart that you'd use to find the value of n. This is a not a comprehensive list of all materials but a shortened table for simplicity's sake. An extensive table can be found in hydraulics textooks and government websites.
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Rh = the hydraulic radius of the cross-section
Here is the simplest formula for the hydraulic radius of a semicircle.
Rh = Diameter/4
As a summary, let's do a couple of examples together.
Example 1:
After a heavy rain, a canal behind an apartment building is flowing at 3.5ft high and 6ft wide. The canal is made of concrete with steel forms and has a slight slope of .03? What is the discharge for this canal?
Based on the table, the Manning Coefficient for concrete is 0.011.
Calculate the area of the semicircle for this canal. pi x 3.5² ÷ 2 = 19.24 ft²
The hydraulic radius is 6ft/4 = 1.5ft
Plug the numbers into the equation. Q = (1.49/.011)(19.24)(1.5^2/3)(.03^1/2) = 591ft³/sec
Example 2:
A small stream in a city park has a constant flow that is 1.5ft deep and 4ft wide. The bed is made of lead and the slope is a slight 0.002. What is the rate of flow for this channel?
n = .011
Area = pi x 1.5² ÷ 2 = 3.53 ft²
Hydraulic Radius = 4/4 = 1
Q = (1.49/.011)(3.53)(1^2/3)(.002^1/2) = 21.38 ft³/sec
The Slope Area Method is used to calculate how much water flows through a channel at a certain point, which is called discharge. This method utilizes the Manning Formula. In order to use the Manning Formula, one must assume certain characteristics are uniform, which is not very realistic for natural bodies of water, but works well for man-made channels. The formula looks a bit overwhelming, until breaking down each component. You'd need to plug the slope, area, Roughness Coefficient, and hydraulic radius into the Manning Formula to find the answer. As long as you are careful to find each piece of data and input the values correctly, you can calculate the discharge without issue.
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