Introduction to the Manning Equation for Uniform Open Channel Flow Calculations

Written by: Harlan Bengtson • Edited by: Lamar Stonecypher
Updated May 20, 2011

The Manning equation is an empirical equation for uniform open channel flow. It can be used for water flow rate calculations in either man made or natural open channels. Water flow calculation with the Manning equation uses the channel slope, hydraulic radius, and Manning roughness coefficient.

Introduction

A common use of the Manning Equation is for water flow rate calculation in an open channel. It can also be used to calculate values of other uniform open channel flow parameters such as channel slope, Manning roughness coefficient, or normal depth, when the water flow rate through the open channel is known.

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An example set of calculations later in this article includes average flow velocity determination and water flow calculation for a given channel and flow depth. The Manning equation applies to open channel flow in natural channels as well as to man-made channels. For example, river discharge can be related to the depth of water flow and river parameters like slope, width and cross-sectional shape.

Image Credit: geograph.org.uk

Uniform Open Channel Flow

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Uniform open channel flow takes place whenever there's a constant volumetric flow rate of liquid through a section of channel that has a constant bottom slope, constant hydraulic radius (that is constant channel size and shape), and constant channel surface roughness (constant Manning roughness coefficient). Under these conditions, the liquid will flow at a constant depth, often called the normal depth for the given channel and volumetric flow rate. The diagram at the left shows uniform open channel flow in the upstream portion of the channel, followed by a section of non-uniform flow due to a change in channel bottom slope, leading to another stretch of uniform open channel flow. Uniform flow is a necessary condition for the use of the Manning Equation, to be discussed in detail in the next section.

Image Credit: H. Bengtson, Reference #1

The Manning Equation

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

and for S.I. units it is: Q = (1.0/n)A(R^2/3)(S^1/2), Where

Q = volumetric water flow rate passing through the stretch of channel, ft³/sec (m³/s for S.I.)

A = cross-sectional area of flow perpendicular to the flow direction, ft² (m² for S.I.)

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

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

R = hydraulic radius = A/P in ft (m for S.I.) where

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

P = wetted perimeter of cross-sectional flow area, ft (m for S.I.)

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.

In S.I. units this equation becomes: V = (1.0/n)(R^2/3)(S^1/2), with average velocity in m/s.

Note that the Manning Equation is an empirical, dimensional equation. With the constant equal to 1.49 for U.S. units (or 1.0 for S.I. units), all of the parameters must have the units given above for the chosen system of units.

The Manning Roughness Coefficient

The Manning roughness coefficient, n, is an experimentally determined constant.

Manning Roughness Coefficient Table of Values

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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. The table above shows Manning roughness coefficient values for severa surfaces commonly used for open channel flow. In general smoother surfaces have lower Manning roughness coefficient values and rougher surfaces have higher Manning roughness coefficient values.

Table Source: Reference #1

Example Water Flow Rate Calculation for a Rectangular Open Channel

Problem Statement: Consider an open channel of rectangular cross-section, with bottom width of 4

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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?

Solution: The flow velocity and water flow rate calculation can both be made using the Manning Equation in the two forms given in the previous section.

the hydraulic radius is R = A/P = (2)(4)/(4 + 2 + 2) = 1 ft

Substituting values into the velocity form of the Manning equation gives:

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

The open channel water flow rate can then be calculated:

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 unknown. 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. The Manning equation can also be used for experimental determination of the Manning roughness coefficient of a given channel, by measuring all of the other parameters (Q, A, P and S) and calculating the Manning roughness coefficient using the Manning equation.

For downloadable Excel templates that can be used for Manning equation/uniform open channel flow calculations, see the article, "Use of Excel Spreadsheet Formulas for Uniform Open Channel Flow/Manning Equation Calculations."

An article with Excel templates to calculate normal depth for uniform open channel flow is "Calculation of Normal Depth for Open Channel Flow with Excel Formulas."

For information on how to make partially full pipe flow calculations, see: "Use of Excel Spreadsheet Templates for Partially Full Pipe Flow."

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.

References and Image Credits

1. Bengtson, Harlan H., Open Channel Flow I - The Manning Equation and Uniform Flow, an online, continuing education course for PDH credit.

2. U.S. Dept. of the Interior, Bureau of Reclamation, 2001 revised, 1997 third edition, Water Measurement Manual.

3. Chow, V. T., Open Channel Hydraulics, New York: McGraw-Hill, 1959.

Harlan Bengtson Feb 7, 2012 5:43 PM

RE: Introduction to the Manning Equation for Uniform Open Channel Flow Calculations

I'm not sure what you mean by "m"

solome Feb 7, 2012 2:08 PM

if side slope is 1:2 what is m

Harlan Bengtson Dec 21, 2011 9:44 PM

See the articles, "Open Channel Flow Spreadsheets - Critical Depth and Critical Slope," at: http://www.engineeringexcelspr... , and "Calculation of Critical Depth and Critical Slope" at: : http://www.brighthub.com/engin.... You can calculate the average velocity in the channel from Fr = V/(gy)^1/2. Then calculate the flow rate per unit width,q, from Q = V(by) and q = Q/b. y is the depth of flow and b is the channel width. Then you can calculate the critical depth, yc, from yc = (q^2/g)^1/3.

Harlan Bengtson Dec 21, 2011 8:00 PM

See the articles: "Open Channel Flow Spreadsheets - Critical Depth and Critical Slope," at: http://www.engineeringexcelspr... , and "Calculation of Critical Depth and Critical Slope" at: : http://www.brighthub.com/engin.... You can calculate the average velocity in the channel from Fr = V/(gy)^1/2. Then calculate the flow rate per unit width,q, from Q = V(by) and q = Q/b. y is the depth of flow and b is the channel width. Then you can calculate the critical depth, yc, from yc = (q^2/g)^1/3.

kapilkma03 Dec 21, 2011 11:14 AM

froude no of rectangular channel is given and its depth of flow is given how to calculate critcal depth

Harlan Bengtson Mar 13, 2011 9:25 AM

b for trapezoid

lower case b typically refers to the bottom width of a trapezoidal cross-section

Gen Mar 13, 2011 3:37 AM

Question on Open Channel Flows

Suppose I have a trapezoidal cross section.

I am supposed to compute q = Q/b. Which b do I use, the bottom width or the top width? Thanks :)

Harlan Bengtson Feb 27, 2011 9:28 AM

One More Equation

The parameter S is the same for the Chezy Eqn and the Manning Eqn. It is the slope of the hydraulic grade line, which for uniform flow, is also the constant bottom slope of the channel.

With regard to angles in equations, for calculating the value of a trigonometric function, like sin A or cos A, the angle A can be in either radians or degrees, as long as you calculate the sin or cos correctly. If an angle, A, appears in an equation without being the argument of a trig function, like A cos B or A log R, the angle, A, usually must be in radians, but an explanation of the variables in the equation should tell you whether the angle must be in radians or in degrees.

mun-eim asmawil Feb 27, 2011 12:38 AM

one more question

does S in maning and chezy is the same?please i am a little bit confused. I am making program then someone told me that if a unit is degree just convert it to radians then substitute to the equation.

Harlan Bengtson Feb 26, 2011 9:16 PM

Slope in Manning Equation

The parameter, S, in the Manning Equation is the bottom slope in the channel. A convenient way to remember the definition of slope is "rise over run." That is, the bottom slope of a given length of channel is the change in elevation divided by the horizontal length of that section of channel.

45 degrees is very very steep for an open channel bottom slope. If the slope is indeed 45 degrees, then the slope = 1/1 = 1.

For a channel with trapezoidal cross-section, another parameter needed in the calculations is the side slope of the trapezoidal cross-section. 45 degrees would be an appropriate value for the side slope of the channel cross-section.

mun-eim asmawil Feb 26, 2011 8:58 PM

hello

sir can you enlighten me about the meaning of the slope in this equation. For example i am given unit of 45 degrees how could i convert this then use in the equation. TIA.

Harlan Bengtson Feb 25, 2011 10:44 AM

normal flow depth

Yes, normal depth less than critical depth implies supercritical flow.

David Williams Feb 23, 2011 1:18 PM

Normal Flow Depth

Ok, that makes sense. The calculated critical depth for the channel is 31mm so do you agree that all flow profiles in the channel will be steep as yc is bigger than yn?

Harlan Bengtson Feb 23, 2011 12:41 PM

Normal Depth Excel Spreadsheet

Here's my interpretation of the 6 mm normal depth: If the channel is long enough downstream of the broadcrested weir to adjust itself to normal depth, it should flow at a depth of 6 mm, after it reaches uniform flow conditions.

David Williams Feb 23, 2011 12:31 PM

Excel Spreadsheet

I see, thankyou for explaining that. I am attempting to calculate the normal depth of flow in an open channel with the following parameters - b = 100mm, n = 0.009(assumed for a plastic channel), S = 0.0097 and Q = 1.72l/s. The spreadsheet is giving a value y0 = 0.02ft (6mm). This is a laboratory experiment using a broadcrested weir. The depth of flow upstream of the weir at this flow rate was 100mm. I'm a little confused with the 6mm answer.

Harlan Bengtson Feb 23, 2011 11:43 AM

Iterative calculations

Right, I think my previous answer may have still been a bit confusing. Here's another try:

The right column is a function of only yo, so the values won't change when you alter the input values. The values in the right column change only when you change yo values in the left column. Here's an example:

I entered the following inputs: b = 4 ft, n = 0.01, S = 0.0003, Q - 20 cfs. This gave Q*n(1.49*S^1/2 = 8.525.

At this point none of the values in the iterative solution columns has changed. By looking at the first three entries in the iterative solution table, I see that yo must be between 2 ft and 3 ft, because the target value of 8.525 is between 5.414 and 9.000. Trying 2.5 for yo gives 7.814 in the right column, so I see that yo is between 2.5 and 3. entering 2.8 and 2.9 for yo shows that yo is between 2.8 and 2.9. Finally, I find that yo = 2.87 gives 8.524 in the right column which is as close as I can get with 3 significant figures for yo.

David Williams Feb 23, 2011 10:48 AM

Excel Spreadsheets

I have entered new values for b, n, S and Q, and have thus obtained a new value for Q*n/(1.49*S1/2) =, but when I enter unknowns in the left hand column of the table, the right hand column is still showing calcs for the example parameters, not the ones I have just entered.

Harlan Bengtson Feb 23, 2011 9:46 AM

Using Excel Spreadsheets

Any of the Excel spreadsheets I've posted that use iterative calculations, require manual input of values for the unknown that your trying to calculate (e.g. normal depth). The spreadsheet will calculate the target value for the parameter in the right column of the part of the spreadsheet set up for the iterative calculations. You need to enter values for the unknown in the first two cells of the left column. (I usually start with 1 and 2.) From the first two values in the right column, you can see whether you should choose a value larger than 2 or less than 1 for your next guess. You continue in this manner until you "zero in on" a value for the unknown that makes the value in the right column equal to or very close to the target value calculated by the spreadsheet.

David Williams Feb 23, 2011 9:17 AM

Using excel spreadsheet

I have entered all the values in to the spreadsheet yet the iterative process at the end remains the same as in the example. What am I doing wrong/not doing?

Harlan Bengtson Feb 22, 2011 10:46 PM

Afflux Effect

see the comment on Afflux over a weir (about 8 comments down)

Jinal Feb 22, 2011 10:34 PM

Afflux effect

I want to know how to calculate the water levels in upstream of the weir. I want to know the effect of afflux in upstream reach (upto 1000km from weir) due to weir.

David Williams Feb 22, 2011 11:59 AM

Calculating normal depth

Thanks for the reply. I'm on the other side of the pond but I am sure I will be able to convert from metric to imperial in order to use that spreadsheet.

Harlan Bengtson Feb 22, 2011 11:01 AM

Calculating Normal Depth

The normal depth in a rectangular channel depends upon the channel width, Manning roughness, bottom slope and flow rate. If you know these four parameters you can calculate it. See the article, "Calculation of Normal Depth for Open Channel Flow with Excel Formulas," at: http://www.brighthub.com/engineering/civil/articles/86170.aspx, for help with the calculations. That article also has information for calculating normal depth in trapezoidal channels.

David Williams Feb 22, 2011 9:48 AM

calculating normal depth

How do I calculate normal depth in an open channel with a gradually varied flow. The channel contains a B.C.Weir and I know critical depth, I now need to calc normal depth so that I can classify surface profiles. This is a laboratory experiment.

Harlan Bengtson Feb 19, 2011 10:10 PM

energy losses

A couple of primary factors affecting energy loss in open channel flow are the roughness of the channel surface and the velocity of the water flow.

fiffy Feb 19, 2011 9:30 PM

open channel

what are factors that contribute the energy losses in open channel

regards
fiffu.

Harlan Bengtson Oct 10, 2010 11:19 AM

Factors that affect discharge in trapezoidal channel

The factors that affect the discharge in a trapezoidal open channel are the depth of flow, bottom width, side slope, bottom slope and Manning roughness. See also two other articles in this series, "Calculation of Hydraulic Radius for Uniform Open Channel Flow," and "Use of Excel Spreadsheet Formulas for Uniform Open Channel Flow/Manning Equation Calculations." You can click on these titles in the list above.

rampradeep Oct 10, 2010 8:36 AM

penman method

what are the parameters which influence the discharge calculation in trapezoidal open channel

Harlan Bengtson Aug 29, 2010 3:52 PM

Afflux of water over a weir

Afflux is the difference in water surface elevation upstream and downstream of an obstruction. In this case the obstruction you are concerned about is a weir. The upstream water surface elevation can be calculated as the head over the weir plus the height of the wier crest above the channel bottom. The downstream water surface elevation will be the normal depth, which can be calculated if you have adequate information about the channel and the flowrate.

See the following three articles for equations relating head over a weir to flow rate over the weir.

"Open Channel Flow Measurement 4: the V-Notch Weir," at http://www2.brighthub.com/adrp2d/engineering/civil/65701.html

"Open Channel Flow Measurement 5: the rectangular weir," at http://www.brighthub.com/engineering/civil/articles/65880.aspx

"Open Channel Flow Measurement 3: the Broad-Crested Weir," at http://www.brighthub.com/engineering/civil/articles/61254.aspx

Harlan Bengtson Aug 29, 2010 3:28 PM

Allowable Velocity

If you do a search for "maximum allowable velocity in open channel," you will find open channel design guidelines for several states that give maximum allowable design velocities for a variety of erodible channel linings. I didn't find brick lining in any of them, but did find the following for similar materials:
Hard cemented conglomerate - max of 8 ft/sec; soft rock - max of 5 - 9 ft/sec; hard rock - 10 - 15 ft/sec.

azam Aug 29, 2010 10:31 AM

Afflux

How afflux of water over a weir is calculated?

azam Aug 29, 2010 10:24 AM

allowable velocity

what is max. allowable velocity for brick lining in an open channel????

regards
azam

Harlan Bengtson Aug 27, 2010 12:46 PM

Flow Depth

Aftab,

In answer to: "In designing a straight wall channel how to calculate normal depth and water depth if manning n is .01 and the slope is .02. Using trail and error method normal depth is calculated but how to calculate water depth?"

If it is uniform flow (as it must be in order to calculate the normal depth using the Manning Equation), then the water depth will be the normal depth.

Harlan Bengtson Aug 27, 2010 12:42 PM

Wetted Perimeter for wide channel divided into parts

If a wide channel is divided into several parts, the wetted perimeter for each part will be the length of channel surface in contact with water for each part of the total cross-section.

Harlan Bengtson Aug 27, 2010 12:39 PM

Software for Manning Equation Calculations

See the article, "Use of Excel Spreadsheet Formulas for Uniform Open Channel Flow/Manning Equation Calculations," at http://www.brighthub.com/engineering/civil/articles/82736.aspx, for downloadable Excel templates for several types of Manning Equation/Uniform Open Channel Flow calculations.

aftab Aug 27, 2010 12:29 PM

flow depth

In designing a straight wall channel how to calculate normal depth and water depth if manning n is .01 and the slope is .02. Using trail and error method normal depth is calculated but how to calculate water depth?

abdo Aug 7, 2010 1:33 PM

Hydrology

thank you very much
please I want software program use to calculate discharge by using manning equation

azam Jul 29, 2010 12:58 AM

velocity profile

I need to know what would be the velocity profile along the side of a open channel
HOw it is calculated??

lloyd Jul 11, 2010 2:32 PM

good day sir. thank you so much for your quick reply on my last email.can you help me with this problem?

The continuity and momentum equation of a 1-D open channel flow in a rectangular channel can be expressed respectively as follows:

(dh/dt)+(dq/dx)=0 eq.(1)

(dq/dt)+(dvq/dx)+(gh(dh/dx))-ghio+((gn^2q^2)/h^7/3)=0 eq. (2)
where h is the water depth (m), q is the unit discharge (m2/s), v is the average velocity (m/s), i0 is the channel slope, and n is mannings rougness coef.

regarding Eq. (2), the firts term is the temporal change of q, while the second term is the acceleration due to discharge change in the x-direction. the 3rd as well as the 4th terms compose the water slope and the 5th the shear stress due to roughness of the river bed.

on the other hand, Manning's Equation, i.e. Eq.(3), can be derived from Eq.(2) as:

q=(1/n)(h^5/3)(i0-dh/dx)^1/2

QUESTION: derive Eq.(3) from Eq.(2) while showing a series of assumptions.

Abiy May 10, 2010 11:40 AM

Cross-sectional sideslopes for a trapezoidal natural river flow

Are there any typical cross-sectional sideslope values for a natural river (with an approximately trapezoidal cross-section)?
thanks!

puji harsanto Apr 18, 2010 8:50 PM

ask wetted perimeter

If I devide wide (transverse direction) channel in many pieces, how to calculate wetted perimeter each part?

Harlan Bengtson Apr 8, 2010 9:44 PM

How to Estimate Mean Flow Depth

A value for river flow rate and a channel profile drawing are not enough information to determine the mean flow depth. There are still two unknowns (flow depth and velocity) and only one equation, V = Q/A. If you can obtain a value for bottom slope of the reach of river channel of interest and an estimate of the manning roughness coefficient for the channel, then you should be able to calculate the flow depth (normal depth) from the Manning equation [Q = (1.49/n)A(Rh^2/3)(S^1/2)], if you can express the river cross-sectional area and hydraulic radius as functions of depth from the channel profile. The solution for normal depth will probably be a trial and error solution.

Harlan Bengtson

Shaki Dada Apr 8, 2010 10:14 AM

How to estimate mean flow depth

How would one estimate the mean water depth if flow rate is known and one has to calculate flow velocity?
I know flow velocity = flow rate/cross-sectional area.
I'm not sure what flow depth to assume in order to calculate flow area.
All the info i have on the river is a profile drawing and flow rate.

Harlan Bengtson Apr 8, 2010 9:26 AM

bottom slope

The bottom slope of an open channel is calculated as the change in elevation of the channel bottom divided by the distance over which that change in elevation takes place. It is usually given as a dimensionless number (ft/ft or m/m, etc). So for example, if the elevation of the channel bottom drops by 25 ft over 1 mile, then its slope is S = 25 ft/5280 ft = 0.00473.

Harlan Bengtson

gourav Apr 8, 2010 7:00 AM

How to calculate the bottom slope of rectangular open channel flow

David Adkins Mar 3, 2010 11:58 AM

The zero slope trough

Thank you for the help

Harlan Bengtson Mar 3, 2010 9:43 AM

The Manning equation will not work for a zero slope trough with a drain in the middle. I assume that flow comes in at one or both ends. The driving force for flow from an end of the trough to the drain will be a difference in elevation of the water surface. When water first enters the end of the channel, it will flow toward the center drain to even out the water level. When water reaches the drain, some will flow down the drain and create a drawdown near the drain. That drawdown will cause water to continue to flow toward the drain because the difference in elevation of the water surface. I believe the best way to make calculations about this flow would be with the energy equation and the continuity equation written between a point at the entry to the trough and a point at the drain. The frictional head loss would be hardest part to handle. The frictional head loss should be approximately proportional to the velocity squared, so at low flows, the frictional loss will be low, but at high flows the frictional head loss could be significant.

David Adkins Mar 1, 2010 7:01 PM

Open channel flow in a level trough w drain

How would you model open channel flow in a trough that has zero slope in the longitudinal direction and a drain located in the center? Length is about 275 feet. Surface is asphalt.

Thanks,
David Adkins

Harlan Bengtson Feb 13, 2010 9:38 AM

100mm wide by 50 mm deep flow

For 50 mm deep flow of water in a 100 mm wide channel, A = (50)(100) sq mm = 5000 sq mm = 0.005 sq m.

The wetted perimeter is P = 2(50) + 100 mm = 200 mm = 0.2 m

The hydraulic radius is Rh = A/P = 0.005/0.2 = 0.025 m

You still need to know the channel slope, S, and the Manning Roughness, n, in order to calculate channel flow rate using the Manning Eqn (in S.I. units): Q = (1/n)(A)(Rh^2/3)(S^1/2)

j.venkatesan Feb 13, 2010 4:24 AM

how to calculate in channel in with100/height 50mm

channel calculate in 100mm/50mm

Introduction to the Manning Equation for Uniform Open Channel Flow Calculations

Introduction

Uniform Open Channel Flow

The Manning Equation

The Manning Roughness Coefficient

Example Water Flow Rate Calculation for a Rectangular Open Channel

Summary

References and Image Credits

Comments