The earliest water quality models of distribution systems were steady-state models (Wood, 1980 and Males, Clark, Wehrman, and Gates, 1985). These models used simultaneous equation or “marching out” solution methods to determine the steady- state water quality concentrations throughout the distribution systems. However, it quickly became apparent that steady-state water quality models were of limited use in representing actual systems due to the temporal variability in distribution system operation, the impacts of tanks on water quality, and temporal changes in source con- centrations. This led to the development of several dynamic water quality models dur- ing the mid to late 1980s (Clark, Grayman, Males, and Coyle, 1986; Hart, Meader, and Chiang, 1986; Liou and Kroon, 1987; and Grayman, Clark, and Males, 1988). Two methods are available to solve the dynamic water quality equations used in water quality models. One method is based on an Eulerian approach that divides each sepa- rate pipe into a series of equal length sub-links. The other method is a Lagrangian approach that tracks parcels of water of homogeneous water quality concentrations as they move through the pipe system. These solution methods are shown graphically in Figures 2.27 and 2.28 and explained in detail in the paragraphs that follow. In both solution methods, a hydraulic model must first be applied in extended-period simula- tion (EPS) mode to determine the flow, flow direction, and velocity in each pipe at all times during the simulation.
The Eulerian Approach. With an Eulerian approach (illustrated in Figure
2.27), an observer located at a fixed location watches water as it flows by. Grayman, Clark, and Males (1988) developed an Eulerian solution method for water quality modeling in distribution systems, and Rossman, Boulos, and Altman (1993) formal- ized this method and named it the Discrete Volume Method (DVM).
In DVM, for each time period, a pipe is divided into a series of sub-links with the sub- link length selected so that the time of travel through each sub-link is equal to a user- selected water quality time step that remains constant throughout the simulation. As a result, water moves from one sub-link to the next adjacent downstream sub-link in one water quality time step.
In order to meet this constraint, the sub-link length varies from pipe to pipe and within a pipe as flow changes. If the constituent being simulated is reactive, then the water quality concentration is adjusted according to the appropriate reaction method during each water quality time step.
At a junction at the downstream end of one or more pipes, the water quality concen- tration in the junction is calculated by taking a flow-weighted average of incoming inflows, as described previously by Equation 2.42. Water moves instantaneously through pumps and valves without a change in water quality. At the end of a hydraulic time step, if there is a change in flow or direction, then the sub-link gridding is changed, and the water quality concentrations at the end of the previous time step are used to define the initial water quality in each of the new sub-links. There are special numerical assumptions made to accommodate “problem” situations such as very short pipes (with travel time less than the water quality time step) and very long pipes (with a very large number of sub-links).
Figure 2.27
Eulerian solution method
The Lagrangian Approach. In the Lagrangian approach (illustrated in Figure
2.28), rather than observing the flow from the “sidelines,” the observer moves with the flow. Additionally, rather than having a fixed grid, parcels of water with homoge- neous concentrations are tracked through the pipe. New parcels are added when water quality changes occur due to changes in source quality or when parcels are combined at junctions. In order to reduce the number of parcels, algorithms have been devel- oped for combining adjacent parcels when the difference in concentrations are less than a user-defined tolerance. Liou and Kroon (1987) and Hart, Meader, and Chiang (1986) developed Lagrangian solution methods for water distribution system water quality models.
Lagrangian solutions can be either time-driven or event-driven. In a time-driven method, conditions are updated at a fixed time step. In an event-driven model, condi- tions are updated when the source water quality changes or when the front of a parcel
References 65
reaches a junction. In comparing the methods, Rossman and Boulos (1996) found that the Lagrangian time-driven method is the most efficient and versatile of the solution methods for water distribution system quality models.
Figure 2.28
Lagrangian solution method
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