Hydraulic Radius for Uniform Open Channel Flow

Introduction

One of the parameters needed in order to make use of the Manning equation for open channel flow calculations is the hydraulic radius of the channel cross section. Common shapes for open channel cross section include rectangle, trapezoid, triangle and circle.

The use of hydraulic radius in Manning equation calculations is covered in the first article of this series, 'Introduction to the Manning Equation for Open Channel Flow Calculations' and the hydraulic radius and use of the Manning equation for a circular pipe are covered in 'How to Use the Manning Equation for Storm Sewer Calculations.'

This article will cover rectangle, trapezoid and triangle shapes for an open channel cross section. Hydraulic radius is defined as the cross sectional area of flow divided by the wetted perimeter, so the calculation of rectangle and trapezoid area and triangle area will be included along with the perimeter for each.

Rectangular Cross Section

The simplest open channel flow cross section for calculation of hydraulic radius is a rectangle. The depth of flow is often represented by the symbol, y, and b is often used for the channel bottom width, as shown in the diagram at the left. From the hydraulic radius definition: R_H = A/P, where A is the cross sectional area of flow and P is its wetted perimeter. From the diagram it is clear that A = by and P = 2y + b, so the hydraulic radius is: R_H = by/(2y + b) for an open channel flow through a rectangular cross section.

Trapezoidal Cross Section

A trapezoid shape is sometimes used for man made channels and the cross section of natural stream channels are often approximated by a trapezoid area. The diagram at the right shows a trapezoid and the parameters typically used for its shape and size in open channel flow calculations. Those parameters, which are used to calculate the trapezoid area and wetted perimeter, are y, the liquid depth; b, the bottom width; B the width of the liquid surface; λ, the wetted length measured along the sloped side; and α, the angle of the sloped side from vertical. The side slope is usually specified as horiz:vert = z:1.

The cross sectional area of flow is the trapezoid area: A = y(b + B)/2, or

A = (y/2)(b + b + 2zy), because B = b + 2zy, as can be seen from the diagram.

Simplifying, the trapezoid area is: A = by + zy².

The wetted perimeter is: P = b + 2λ, but by Pythagoras Theorem:

λ² = y² + (yz)², or λ = [y² + (yz)²]^1/2, so the wetted perimeter is:

P = b + 2y(1 + z²)^1/2, and the hydraulic radius for a trapezoid is:

R_H = (by + zy²)/[b + 2y(1 + z²)^1/2]

Triangular Cross Section

A triangular open channel cross section is shown in the diagram at the left. The diagram shows the typical case, where the two sides are sloped at the same angle.There are less parameters needed for the triangule area than for the trapezoid area. The parameters, as shown in the diagram are: B, the surface width of the liquid; λ, the sloped length of the triangle side; y, the liquid depth measured from the vertex of the triangle; and the side slope specification, horiz:vert = z:1.

The triangle area is: A = By/2, but the figure shows that B = 2yz, so the triangle area becomes simply: A = y²z.

The wetted perimeter is: P = 2λ with λ² = y² + (yz)². This simplifies to: P = 2[y²(1 + z²)]^1/2

The hydraulic radius is thus: RH = A/P = y²z/{2[y²(1 + z²)]^1/2}

About the Author

Dr. Harlan Bengtson is a registered professional engineer with 30 years of university teaching experience in engineering science and civil engineering. He holds a PhD in Chemical Engineering.

Related Reading

Open Channel Flow Measurement 1: Introduction to the Weir and Flume - Open channel flow measurement is usually done with a weir or a flume. The weir or flume causes a change in water depth that correlates with water flow rate. Common open channel flow meters are the sharp crested weir (v notch weir and rectangular weir), broad crested weir, and Parshall flume

Open Channel Flow Basics 2: Supercritical Flow - Supercritical flow is open channel flow with high flow velocity and depth less than critical depth. Subcritical flow has a low flow velocity and depth that is deeper than critical depth. The Froude number will be greater than one for supecritical flow and less than one for subcritical flow.

Uniform Open Channel Flow and the Manning Equation

The Manning equation is widely used for uniform open channel flow calculations with natural or man made channels. The Manning equation is used to relate parameters like river discharge and water flow velocity to hydraulic radius, and open channel slope, size, shape, and Manning roughness.

• 1. Introduction to the Manning Equation for Uniform Open Channel Flow Calculations
• 2. Calculation of Hydraulic Radius for Uniform Open Channel Flow
• 3. Use of the Manning Equation for Open Channel Flow in Natural Channels

Read more: http://www.brighthub.com/engineering/civil/articles/67126.aspx#ixzz0jSIYUe8z


1. Introduction to the Manning Equation for Uniform Open Channel Flow Calculations

Introduction

Use of the Manning Equation to calculate the values of uniform open channel flow parameters such as channel slope, Manning roughness coefficient, water flow rate or flow velocity will be presented. An example calculating water flow rate and average flow velocity for a given channel and flow depth is included. The Manning equation applies to open channel flow in natural channels as well. For example, river discharge can be related to the depth of water flow and river parameters like slope, width and cross-sectional shape.

Uniform Open Channel Flow

Uniform open channel flow takes place whenever there is a constant volumetric flow rate of liquid through a section of channel that has a constant bottom slope, constant channel size and shape, and constant channel surface roughness. Under these conditions, the liquid will flow at a constant depth, often called the normal depth for the given channel and volumetric flow rate. Uniform flow is a necessary condition for the use of the Manning Equation, the primary topic of this article. Open channel flow can take place in man made or natural channels. In a natural channel, river discharge is often a parameter of interest.

The Manning Equation

The Manning Equation for U.S. units is: Q = (1.49/n)A(R^2/3)(S^1/2), Where

Q = volumetric water flow rate passing through the stretch of channel, ft³/sec,

A = cross-sectional area of flow perpendicular to the flow direction, ft²,

S = bottom slope of channel, ft/ft (dimensionless),

n = Manning rougness coefficient (empirical constant), dimensionless,

R = hydraulic radius = A/P, where

A = cross-sectional area of flow as defined above,

P = wetted perimeter of cross-sectional flow area, ft.

The Manning Equation can be expressed in terms of flow velocity instead of flow rate. Using the equation, V = Q/A as a definition for average flow velocity, the Manning Equation becomes:

V = (1.49/n)(R^2/3)(S^1/2), with average flow velocity in ft/sec.

Note that the Manning Equation is an empirical, dimensional equation. With the constant equal to 1.49, all of the parameters must have the units given above.

The Manning Roughness Coefficient

The Manning roughness coefficient, n, is an experimentally determined constant. Its value depends upon the nature of the channel and its surface. Tables giving values of n for different man-made and natural channel types and surfaces are available in many textbooks, handbooks and on-line. Here are a few typical values for n:

Brick - n = 0.015, new cast-iron - n = 0.012, concrete - n - 0.011 to 0.015, corrugated metal - n = 0.022

Example Manning Equation Calculation

Consider an open channel of rectangular cross-section, with bottom width of 4 ft, containing water flowing 2 ft deep. The bottom slope of the channel is 0.0004 and it is made of concrete with a Manning roughness coefficient of 0.011. What would be the average flow velocity of the water and what would be the volumetric water flow rate? These can both be calculated using the Manning Equation in the two forms given in the previous section.

Hydraulic radius = R = A/P = (2)(4)/(4 + 2 + 2) = 1 ft

V = (1.49/0.011)(1^2/3)(0.0004^1/2) = 2.71 ft/sec

Q = VA = (2.71 ft/sec)(8 ft²) = 21.7 ft³/sec

Any of the other parameters in the Manning Equation could be calculated if it is the only unkown. For example, the channel bottom slope needed to carry a given flow rate in a channel of given shape and size at a given depth of flow could be calculated.

Summary

The Manning equation is useful for a variety of open channel flow calculations involving parameters such as water flow rate, flow velocity, channel slope, channel roughness, water flow depth, and channel size and shape parameters. For a natural channel, river discharge (water flow rate) is often a parameter to be determined.

About the Author

Dr. Harlan Bengtson is a registered professional engineer with 30 years of university teaching experience in engineering science and civil engineering. He holds a PhD in Chemical Engineering

Related Reading

Open Channel Flow Measurement 1: Introduction to the Weir and Flume - Open channel flow measurement is usually done with a weir or a flume. The weir or flume causes a change in water depth that correlates with water flow rate. Common open channel flow meters are the sharp crested weir (v notch weir and rectangular weir), broad crested weir, and Parshall flume.

How to Use the Manning Equation for Storm Sewer Calculations - Even though storm sewers are circular pipes, the Manning equation can be used for stormwater flow velocity calculation, because the storm sewer is an open channel with the liquid flowing under gravity. Hydraulic radius and pipe roughness coefficient are used in the Manning equation calculations.

Open Channel Flow Basics 2: Supercritical Flow - Supercritical flow is open channel flow with high flow velocity and depth less than critical depth. Subcritical flow has a low flow velocity and depth that is deeper than critical depth. The Froude number will be greater than one for supecritical flow and less than one for subcritical flow.

Uniform Open Channel Flow and the Manning Equation

The Manning equation is widely used for uniform open channel flow calculations with natural or man made channels. The Manning equation is used to relate parameters like river discharge and water flow velocity to hydraulic radius, and open channel slope, size, shape, and Manning roughness.

3. Use of the Manning Equation for Open Channel Flow in Natural Channels

Introduction

The Manning equation can be used for a variety of open channel flow calculations for natural channels, like rivers, streams and canals. Parameters like river discharge, water flow, and flow velocity can be related to slope and size & shape and roughness characteristics of natural channels. For flow in a natural channel, however, the bottom slope and channel size, shape & roughness are less clearly defined and more variable than for water flow in a man made open channel.

The meaning of uniform open channel flow along with details of the Manning equation and the parameters in the equation are presented in the first article of this series, Open Channel Flow Basics I: The Manning Equation and Uniform Flow, so they won't be repeated here.

Uniform Flow in a Natural Open Channel

There must be uniform open channel flow in a natural channel (river, stream, etc) in order to use the Manning equation. That is, the bottom slope, cross-section size, shape & roughness characteristics of natural channels must be at least approximately constant. River channel characteristics are usually not all that uniform over an extended length, but a particular section (called a reach) of a river channel may have reasonably constant slope, size, etc. If that is the case, then the water flow depth and flow velocity will also be constant for that channel reach, and the Manning equation [ Q = (1.49/n)(A)(R_H^2/3)(S^1/2) ] can be used for water flow calculations involving river discharge, Q; cross-sectional area of flow, A; hydraulic radius, R_H; channel bottom slope, S; and Manning roughness coefficient, n. The relationship, V = Q/A can be used to calculate average flow velocity.

Manning Roughness Coefficient Values for Natural Channels

The Manning roughness coefficient, n, is an important parameter for open channel flow calculations with the Manning Equation. The value of n has an effect on variables like river discharge (water flow rate) and flow velocity. A reasonably accurate value of n can be obtained for most man made open channels, but obtaining good values of the Manning roughness coefficient for a reach of natural channel is a bit more of a challenge, because of the greater variability in the nature of the bottom and side surfaces of a natural river channel. Tables with values of n for natural channels are available in many textbooks & handbooks and on the internet. The table in this section is an example from a state agency. It is part of a table from the Indiana Department of Transportation Design Manual. The internet reference is given in the references section at the end of this article.

Example Calculation

Problem Statement: A stream on a plain has a reach that is described as clean and winding with some pools and some weeds. This reach has a reasonably constant slope of 0.0002. The stream cross-section of flow can be approximated as a trapezoid with bottom width equal to 5 feet and side slopes having horiz:vert = 3:1. Use the Manning equation with estimated maximum and minimum values of n for open channel flow in this stream to find the range of river discharge and water flow velocity to be expected for a 3 ft water depth.

Solution: As given above: bottom width, b = 5 ft; channel bottom slope, S = 0.0003; side slope, z = 3; and flow depth, y = 3 ft. From the table in the previous section, for a 'stream on plain, winding with some pools or shoals and some weeds or rocks', the maximum expected value of n is 0.050 and the minimum is 0.35. The second article in this series, 'Calculation of Hydraulic Radius for Open Channel Flow' gives an equation for hydraulic radius of a trapezoidal channel as follows:

R_H = (by + zy²)/[ b + 2y(1 + z²)^1/2 ].

Substituting known values gives: R_H = [ (5)(3) + 3(3)²]/[5 + (2)(3)(1 + 3²)^1/2] = 2.88 ft.

Also, the trapezoid area is A = by + zy² = (5)(3) + 3(3²) = 69 ft².

Values can now be substituted into the Manning equation [Q = (1.49/n)A(R_H^2/3)S^1/2] to give:

For minimum n (0.035): Q_max = (1.49/0.035)(69)(2.88^2/3)(0.0003^1/2) = 103 cfs

For maximum n (0.050): Q_min = (1.49/0.050)(69)(2.88^2/3)(0.0003^1/2) = 72.1 cfs

Using V = Q/A: V_max = 103/69 = 1.49 ft/sec; V_min = 72.1/69 = 1.04 ft/sec

Reference and Image Credit

Indiana Department of Transportation Design Manual: http://www.in.gov/dot/div/contracts/standards/dm/english/Part4/ECh30/ch30.htm

River images: http://www.epa.gov/bioindicators/html/photos_rivers.html

About the Author

Dr. Harlan Bengtson is a registered professional engineer with 30 years of university teaching experience in engineering science and civil engineering. He holds a PhD in Chemical Engineering.

Related Reading

Open Channel Flow Measurement 3: The Broad Crested Weir - The broad crested weir principle of operation is interesting in its use of critical flow over the weir crest. It is more robust than a sharp crested weir and is widely used for open channel flow measurement in natural channels, like rivers and canals. A minimum weir height ensures critical flow.

Hydrology (The Study of Water) 3: Flood Routing - Flood routing is a topic in hydrology used to predict the effect at a downstream location, of a storm in the upstream portion of a river basin. Upstream and downstream hydrographs are related to each other using a flood model. The peak river level can be predicted at downstream locations.

Uniform Open Channel Flow and the Manning Equation

The Manning equation is widely used for uniform open channel flow calculations with natural or man made channels. The Manning equation is used to relate parameters like river discharge and water flow velocity to hydraulic radius, and open channel slope, size, shape, and Manning roughness

Read more: http://www.brighthub.com/engineering/civil/articles/67225.aspx#ixzz0jSJftmQW