City of Morgan Hill Sewer System Master Plan 2017

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:

డ஺ డ௧ ൅ డொ డ௫ ൌ Ͳ (A) (B)

x Continuity Equation:

__

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:

డொ డ௧ ൅ డ డ௧ ቀ ߚ ொ మ ஺ ቁ ൅ ݃ ܣ డ௬ డ௫ ൅ ݃ ܣ ܵ ௙ െ ݃ ܣ ܵ ௢ ൌ Ͳ (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

x Dynamic Equation:

x

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

August 2017

3-2

City of Morgan Hill Sewer System Master Plan