Civil Engineering Reference
In-Depth Information
2
kk
w
ƒ
k
k
(22-6)
b
R
(
k
k
)
w
ƒ
Even if
k
b
and
k
w
where the same throughout a system, the apparent rate
k
could still
vary from one pipe to the next because of variations in pipe size and flow rate.
System of Equations
When applied to a network as a whole, Equations 22-1,
22-2, and 22-3 represent a coupled set of differential / algebraic equations with
time-varying coefficients that must be solved for
C
i
in each pipe
i
and
C
s
in storage
facility
s
. Externally imposed conditions are:
Initial conditions that specify
C
i
in each pipe
i
and
C
s
in each storage facility
s
at time zero
•
Boundary conditions that specify values for external flows and concentrations at
each node
k
which has external mass inputs
•
Hydraulic conditions that specify the volume
V
s
in each storage facility
s
and the
flow
Q
i
in each link
i
at all times
t
•
Dynamic Model Solutions
Although steady-state models are much simpler to set up and solve, their restrictive
assumptions limit their applicability. The development of dynamic models has greatly
advanced the utility of distribution system water quality models. Dynamic models
account for changes in flows through pipes and storage facilities occurring over an
extended period of system operation and how they affect water quality. These models
provide a more realistic picture of system behavior than steady-state models.
Solution methods use one of two approaches for spatial and temporal considera-
tions:
Spatial
Eulerian approaches
divide the pipe network into a series of fixed interconnected
control volumes and record changes at the boundaries or within these volumes as
water flows through them.
•
Lagrangian models
track changes in a series of discrete parcels of water as they
travel through the pipe network.
•
Temporal
•
Time-driven simulations
update the state of the network at fixed time intervals.
Event-driven simulations
update the state of the system only at times when a
change actually occurs, such as when a new parcel of water reaches the end of a
pipe and mixes with water from other connecting pipes.
•
Hydraulic models are used to determine the flow direction and velocity of flow in
each pipe at specific intervals over an extended period. These intervals are referred to
as hydraulic time steps and are typically 1 hour for most applications. Within a hy-
draulic time step, the velocity within each pipe remains constant. Constituent transport
and reaction proceed at smaller intervals of time known as the water-quality time step.
