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Links on this page:  Introduction   Variables   Manning's n coefficients  Error messages   References 

Introduction
The equation beginning V=.... is called the Manning Equation.  It is a semi-empirical equation and is the most commonly used equation for uniform steady state flow of water in open channels (see Discussion and References for Open Channel Flow for further discussion).  Because it is empirical, the Manning equation has inconsistent units which are handled through the conversion factor k.  Uniform means that the water surface has the same slope as the channel bottom.  Uniform flow is actually only achieved in channels that are long and have an unchanging cross-section.  However, the Manning equation is used in other situations despite not strictly achieving these conditions.

In our calculation, most of the combinations of inputs have analytic (closed form) solutions to compute the unknown variables; however, some two require numerical solutions ("Enter Q, n, S, d" and "Enter V, n, S, d").  Our numerical solutions utilize a cubic solver that finds roots of the equations with the result accurate to at least 8 significant digits.  All of our calculations utilize double precision.

It is possible to get two answers using "Enter Q,n,S,d" or "Enter V,n,S,d". This is because maximum Q and V do not occur when the pipe is full. Qmax occurs when y/d=0.938. If y/d is more than that, Q actually decreases due to friction. Given a pipe with diameter d, roughness n, and slope S, let Qo be the discharge when the pipe is flowing full (y/d=1). As seen on the graph below, discharge is also equal to Qo when y/d=0.82. If the entered Q is greater than Qo (but less than Qmax), there will be two solution values of y/d, one between 0.82 and 0.938, and the other between 0.938 and 1. The same argument applies to V, except that Vo occurs at y/d=0.5, and Vmax occurs at y/d=0.81. If the entered V is greater than Vo (but less than Vmax), there will be two solution values of y/d, one between 0.5 and 0.81, and the other between 0.81 and 1.  For further information, see Chow (1959, p. 134).

The following graphs are valid for any roughness (n) and slope (S):
      Qo=full pipe discharge; Vo=full pipe velocity:

Variables                        To top of page
A = Flow cross-sectional area, determined normal (perpendicular) to the bottom surface [L²].
d = Culvert diameter [L].
F = Froude number.  F is a non-dimensional parameter indicating the relative effect of inertial effects to gravity effects.  Flow 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.   Required since the Manning equation is empirical and its units are inconsistent.
n = Manning coefficient.  n is a function of the culvert material, such as plastic, concrete, brick, etc.  Values for n can be found in the table below of Manning's n coefficients.
P = Wetted perimeter [L].  P is the contact length (in the cross-section) between the water and the culvert.
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 culvert [L].  If the culvert has a small slope (S), then entering the vertical depth introduces only minimal error.
Ø = Angle representing how full the culvert is [radians].  A culvert with Ø=0 radians (0^o) contains no water, a culvert with Ø=pi radians (180^o) is half full, and a culvert with Ø=2 pi radians (360^o) is completely full.

Manning's n Coefficients                                To top of page
The table shows the Manning n values for materials most commonly used for culverts.  These values 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).  A more complete table of Manning n values can be found on our Manning n page.

Error Messages                                            To top of page
"Infeasible Input. T/d > 1."  Water top width cannot be greater than the culvert diameter.
"An input is <= 0."  Certain inputs must be positive.
"Infeasible Input. T < 0."  Water top width cannot be negative.
"Infeasible Input. y/d > 1."  Water depth cannot exceed the pipe diameter.

References (footnotes refer back to Manning n table)                      To top of page
^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.

Chow, V. T.  1959.  Open-Channel Hydraulics.  McGraw-Hill, Inc.

^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.

^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.

© 1998-2000 LMNO Engineering, Research, and Software, Ltd. (All Rights Reserved)
Revision 0 on 12/17/1998.  Revision 1 on 7/13/2000 (additional units).
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