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238 Chapter 7 Water Distribution Systems: Modeling and Computer Applications
To overcome the problem of unknown initial conditions at the vast majority of loca-
tions within the water distribution model, the duration of the analysis must be long enough
for the system to reach equilibrium conditions. Note that a constant value does not have to
be reached for equilibrium to be achieved; rather, equilibrium conditions are reached when
a repeating pattern in age, trace, or constituent concentration is established.
Pipes usually reach equilibrium conditions in a short time, but storage tanks are much
slower to show a repeating pattern. For this reason, extra care must be taken when setting a
tank’s initial conditions, in order to ensure the model’s accuracy.

7.6.5 Numerical Methods
Several theoretical approaches are available for solving water quality models. These meth-
ods can generally be grouped as either Eulerian or Lagrangian in nature, depending on the
volumetric control approach that is taken. Eulerian models divide the system into fixed
pipe segments, and then track the changes that occur as water flows through these seg-
ments. Lagrangian models also break the system into control volumes, but then track these
water volumes as they travel through the system. This chapter presents two alternative ap-
proaches for performing water quality constituent analyses.

7.6.6 Discrete Volume Method
The discrete volume method (DVM) is a Eulerian approach that divides each pipe into
equal segments with completely mixed volumes (Fig. 7.5). Reactions are calculated within
each segment, and the constituents are then transferred to the adjacent downstream seg-
ment. At nodes, mass and flow entering from all connecting pipes are combined (assuming
total mixing). The resulting concentration is then transported to all adjacent downstream
pipe segments. This process is repeated for each water quality time step until a different
hydraulic condition is encountered. When this occurs, the pipes are divided again under the
new hydraulic conditions, and the process continues.

7.6.7 Time-Driven Method
The time-driven method (TDM) is an example of a Lagrangian approach (Fig. 7.6). This
method also breaks the system into segments, but rather than using fixed control volumes

Original concentrations

Reacted concentrations

Transport into nodes

Transport into links

Figure 7.5 Eulerian Discrete Volume Method (DVM)