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August 2017
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City of Morgan Hill
Sewer System Master Plan
St. Venant’s Equation for Pipe Capacity
Dynamic modeling facilitates the analysis of unsteady and non-uniform flows (dynamic flows)
within a sewer system. Some hydraulic modeling programs have the ability to analyze these types
of flows using the St. Venant equation, which take into account unsteady and non-uniform
conditions that occur over changes in time and cross-section within system pipes.
The St. Venant equation is a set of two equations, a continuity equation and a dynamic equation,
that are used to analyze dynamic flows within a system. The first equation, the continuity
equation, relates the continuity of flow mass within the system pipes in terms of: (A) the change in
the cross-sectional area of flow at a point over time and (B) The change of flow over the distance
of piping in the system. The continuity equation is provided as follows:
x
Continuity Equation:
డ డ௧
డொ డ௫
ൌ Ͳ
(A) (B)
__
Where:
t = time
x = distance along the longitudinal direction of the channel
Q = discharge flow
A = flow cross-sectional area perpendicular to the x directional axis
The second equation, the dynamic equation, relates changes in flow to fluid momentum in the
system using: (A) Changes in acceleration at a point over time, (B) Changes in convective flow
acceleration, (C) Changes in momentum due to fluid pressure at a given point, (D) Changes in
momentum from the friction slope of the pipe and (E) Fluid momentum provided by gravitational
forces. The dynamic equation is provided as follows:
x
Dynamic Equation:
డொ డ௧
డ డ௧
ቀ
ߚ
ொ
మ
ቁ ݃
ܣ
డ௬ డ௫
݃
ܣ
ܵ
െ ݃
ܣ
ܵ
ൌ Ͳ
x
(A) (B) (C) (D) (E)
__
Where:
t = time
x = distance along the longitudinal direction of the channel
Q = discharge flow
A = flow cross-sectional area perpendicular to the x directional axis
y = flow depth measured from the channel bottom and normal to the x
directional axis
S
f
= friction slope
S
o
= channel slope
β = momentum
g = gravitational acceleration
Use of this method of analysis provides a more accurate and precise analysis of flow conditions
within the system compared to steady state flow analysis methods. It must be noted that two