Trapezoidal Open Channel Design Calculation

Uses Manning Equation.  Compute velocity, discharge, depth, top width, bottom width, area, wetted perimeter, hydraulic radius, Froude number, Manning coefficient, channel slope.

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Gradually varied flow in trapezoidal channel
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Links on this page: Introduction  Variables  Manning n coefficients  Error Messages  References

Introduction
Many natural and man-made channels are approximately trapezoidal.  This calculation uses the most commonly used equation for analyzing open channels - the Manning equation.   It is the equation beginning with V=...  above.  The Manning equation is best used for uniform steady state flows.  Uniform means that the cross-section geometry of the channel remains constant along the length of the channel, and steady state means that the velocity, discharge, and depth do not change with time.  Though these assumptions are rarely ever strictly achieved in reality, the Manning equation is still used to model most open channel flows where conditions are relatively steady and for reaches (portions of rivers) that have a reasonably constant cross-section for a long enough distance that the depth remains fairly constant.

The Manning equation is a semi-empirical equation.  Thus, its units are inconsistent.  The factor, k, has units which allow the equation to be used properly.   Our calculation takes care of all the unit conversions for you, and allows you to enter and compute variables in a wide variety of units.

In our calculation, most of the combinations of inputs have analytic (closed form) solutions to compute the unknown variables; however, some require numerical solution.  Our numerical solutions utilize a cubic solver that finds roots of the equations with the result accurate generally to at least 8 significant digits.  All of our calculations utilize double precision.  Two depths and bottom widths are possible for certain combinations of entered values when Q, T, n, and S or V, T, n, and S are entered.

Variables      [] indicates dimensions                               To calculation
A = Flow cross-sectional area, determined normal (perpendicular) to the bottom surface [L2].
b = Channel bottom width [L].
F = Froude number.  F is a non-dimensional parameter indicating the relative effect of inertial effects to gravity effects.  Flows with F<1 are low velocity flows called subcritical.  F>1 are high velocity flows called supercritical.   Subcritical flows are controlled by downstream obstructions while supercritical flows are affected by upstream controls.  F=1 flows are called critical.
g = acceleration due to gravity = 32.174 ft/s2 = 9.8066 m/s2.   g is used in the equation for Froude number.
k = unit conversion factor = 1.49 if English units = 1.0 if metric units.  Our software converts all inputs to SI units (meters and seconds), performs the computations using k=1.0, then converts the computed quantities to units specified by the user.
n = Manning coefficient.  n is a function of the channel material, such as grass, concrete, earth, etc.  Values for n can be found in the table of Manning's n coefficients shown below.
P = Wetted perimeter [L].  P is the contact length between the water and the channel bottom and sides.
Q = Discharge or flowrate [L3/T].
R = Hydraulic radius of the flow cross-section [L].
S = Slope of channel bottom or water surface [L/L].  Vertical distance divided by horizontal distance.
T = Top width of the flowing water [L].
V = Average velocity of the water [L/T].
y = Water depth measured normal (perpendicular) to the bottom of the channel [L].  If the channel has a small slope (S), then using the vertical depth introduces only minimal error.
z1, z2 = Side slopes of each bank of the channel.  These slopes are computed as horizontal distance divided by vertical distance.
Ø = Angle formed by S.

Manning's n Coefficients                                    To calculation
The Manning's n coefficients were compiled from the references listed under Discussion and References and in the references at the bottom of this web page (note the footnotes which refer to specific references).

Material Manning n Material Manning n
Natural Streams Excavated Earth Channels
Clean and Straight 0.030 Clean 0.022
Major Rivers 0.035 Gravelly 0.025
Sluggish with Deep Pools 0.040 Weedy 0.030
    Stony, Cobbles 0.035
 
Metals Floodplains
Brass 0.011 Pasture, Farmland 0.035
Cast Iron 0.013 Light Brush 0.050
Smooth Steel 0.012 Heavy Brush 0.075
Corrugated Metal 0.022 Trees 0.15
 
Non-Metals
Glass 0.010 Finished Concrete 0.012
Clay Tile 0.014 Unfinished Concrete 0.014
Brickwork 0.015 Gravel 0.029
Asphalt 0.016 Earth 0.025
Masonry 0.025 Planed Wood 0.012
    Unplaned Wood 0.013
Corrugated Polyethylene (PE) with smooth inner walls a,b 0.009-0.015
Corrugated Polyethylene (PE) with corrugated inner walls c 0.018-0.025
Polyvinyl Chloride (PVC) with smooth inner walls d,e 0.009-0.011

Error Messages                                 To calculation
"Invalid boxes checked."  This message is displayed if too many or too few variables are selected to be entered.  A problem cannot be over-stated or under-stated.  In all, there are 30 combinations of inputs which are acceptable.  This message can be displayed to both registered and non-registered users.

The following messages are displayed if an entered value lies outside the acceptable range for the variable.  These messages are only displayed when the "Calculate" button is clicked - for registered users.
"b must be > 0", "b must be 1e-9 to 10000 m", "n must be 1e-9 to 100", "Q must be > 0", "Q must be 1e-9 to 1e9 m3/s", "S must be 1e-9 to 1e9", "T must be > 0", "T must be 1e-9 to 10000 m", "V must be > 0", "V must be 1e-9 to 1e9 m/s", "y must be < 0", "y must be 1e-9 to 1e9 m", "z1, z2 must be >=0", "z1, z2 cannot both be 0".  Note that the channel cannot have both z1=0 and z2=0.  However, if you wish to simulate a rectangular channel, you can set one of them to 0.0 and the other to a very small positive number such as 0.00001.   Or, you can use our rectangular open channel calculation.   Triangular channels can be modeled by setting the bottom width, b, to a very small positive number such as 0.001 m or as low as 10-9 m.

The following messages are displayed if the values entered result in an infeasible situation.  For instance, entering certain combinations of values for T, Q, n, and S can result in an impossible flow situation indicated by a negative bottom width or negative depth.  These messages are only displayed when the "Calculate" button is clicked - for registered users.
"b will be < 0", "T will be > 10000 m", "y will be < 0", "y will be <= 0", "y or b will be < 0".   The calculation uses an upper limit of 10,000 m for top width.

References                                 To calculation
A further discussion of open channel flow, Manning's equation, and trapezoidal channel geometry can be found in these references and on our discussion page.  The Manning's n coefficients shown above are compiled from the references shown here.  The footnotes refer to specific values shown in the Manning n table above.

a Barfuss, Steven and J. Paul Tullis.  Friction factor test on high density polyethylene pipe.  Hydraulics Report No. 208.  Utah Water Research Laboratory, Utah State University.  Logan, Utah.  1988.

c Barfuss, Steven and J. Paul Tullis.  Friction factor test on high density polyethylene pipe.  Hydraulics Report No. 208.  Utah Water Research Laboratory, Utah State University.  Logan, Utah.  1994.

e Bishop, R.R. and R.W. Jeppson.  Hydraulic characteristics of PVC sewer pipe in sanitary sewers.  Utah State University.  Logan, Utah.  September 1975.

Chaudhry, M. H.  1993.  Open Channel Flow.  Prentice-Hall, Inc.

Chow, V. T.  1959.  Open Channel Hydraulics.  McGraw-Hill, Inc. (the classic text)

French, R. H.  1985.  Open Channel Hydraulics.  McGraw-Hill Book Co.

Mays, L. W. editor.  1999.  Hydraulic design handbook.  McGraw-Hill Book Co.

Munson, B.R., D. F. Young, and T. H. Okiishi.  1998.  Fundamentals of Fluid Mechanics.  John Wiley and Sons, Inc.  3ed.

d Neale, L.C. and R.E. Price.  Flow characteristics of PVC sewer pipe.   Journal of the Sanitary Engineering Division, Div. Proc 90SA3, ASCE.  pp. 109-129.  1964.

Streeter, V. L., E. B. Wylie, and K. W. Bedford.  1998.  WCB/McGraw-Hill. 9ed.

b Tullis, J. Paul, R.K. Watkins, and S. L. Barfuss.  Innovative new drainage pipe.  Proceedings of the International Conference on Pipeline Design and Installation, ASCE.  March 25-27, 1990.

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