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556 Chapter 15 Sewerage Systems: Modeling and Computer Applications
methods to determine the shape and distribution of a hydrograph routed through a gravity
pipe: (a) convex routing and (b) weighted translation routing.

15.1.1 Convex Routing
The underlying assumptions of the convex routing method are that the routed outflow for a
time step is based on the inflow and outflow for the previous time step, and that the flow
does not back up in the pipe (that is, no reverse flow or reduced flow due to tailwater ef-
fects exists). Each outflow ordinate is calculated as follows:
O t+¢t = cI + (1 - c)O t (15.1)
t
where
O t t outflow at time t t (L/s, gpm)
t current time (s)
t hydrologic time step (s)
c convex routing coefficient
I t inflow at time t (L/s, gpm)
O t outflow at time t (L/s, gpm).
The convex routing coefficient is essentially a ratio of the hydrologic time step and
representative flow travel time through the pipe, and is calculated as follows:
v ¢t
c =¢t = (15.2)
L t t
where
t hydrologic time step (s)
t t travel time (s)
v velocity established for representative flow (m/s, ft/s)
L length of pipe (m, ft).
In SewerCAD, the velocity used to calculate the coefficient is either the normal veloc-
ity or full velocity generated for a user-specified percentage of the peak of the inflow hy-
drograph. In other words, if the percentage of the peak flow is greater than the capacity of
the pipe, the full-flow velocity is used. If the percentage of the peak flow is less than the
pipe capacity, the flow velocity for normal depth is used.
The higher the percentage of flow, the faster the velocity used to calculate the convex
routing coefficient, and the closer the routed hydrograph will be to a pure translation of the
inflow hydrograph.
The user-specified percentage can be modified in the calculation options. A typical
value is around 75%, but this value may be modified for oddly shaped hydrographs with
sharp, uncharacteristic peaks or for calibration purposes.

15.1.2 Weighted Translation Routing
The convex routing method is only valid when the convex routing coefficient, c, is less
than 1 or when the hydrologic time step is less than the calculated travel time. For certain
cases in which the travel time exceeds the hydrologic time step, SewerCAD automatically
uses an alternative method of routing called weighted translation routing.
Each ordinate of the outflow hydrograph is derived from a weighted average of the or-
dinates for the current and previous time steps of the inflow hydrograph. The weights are
calculated based on the convex routing coefficient.