Hydraulic Design of Highway Culverts - DOT On-Line Publications

B. Methodology
The mathematical solution of the preceding situation is referred to as a storage routing problem.
Conservation of mass, as defined in the Continuity Equation, is essential in formulating the
solution. Simply stated, the rate of change in storage is equal to the inflow minus the outflow.
In differential form, the equation may be expressed as follows:
ds / dt = I − O
(13)
ds/dt is the rate of change of storage
I is the rate of inflow
O is the rate of outflow
An acceptable solution may be formulated using discrete time steps (∆t). Equation (13) may be
restated in this manner:
( ∆s / ∆t) i j = I - O (14)
I and O equal the average rates of inflow and outflow for the time step ∆t from time i to time j.
By assuming linearity of flow across a small time increment, the change of storage is expressed
as:
⎡⎛
Ii + Ij
⎞ ⎛ Oi
+ O j ⎞⎤
⎢⎜
⎟ ⎜ ⎟
⎜
⎥ ∆t
= ∆s
⎢⎣
2 ⎟
−
⎜ 2 ⎟
⎝ ⎠ ⎝ ⎠⎥⎦
"I" and "j" represent the time at the beginning and end of the time increment ∆t.
Figure V-3 depicts an increment of storage across a typical time increment. Note: the smaller
the time increment, the better the assumption of linearity of flows across the time increment.
There are two unknowns represented in Equation (15); therefore, the equation cannot be solved
directly. The two unknowns are the increment of storage, ∆s, and the outflow at the end of the
time increment, Oj. Given a design inflow hydrograph, the known values include each inflow
value, the time step which is selected, and the outflow at the beginning of the time step solved
for during the previous time step. Equation (15) can be rewritten as:
I + I + ( 2s
/ ∆t
− O)
= ( 2s
/ ∆t
+ O)
(16)
i
j
i
j
where the two unknowns are grouped together on the right side of the equality. Because an
equation cannot be solved with two unknowns, it is desirable to devise another equation with the
same two unknowns. In this case, a relationship between storage and outflow is required.
Since both storage and outflow can be related to water surface elevation, they can be related to
one another. This second relationship provides a means for solving the routing equation. The
method of solution is referred to as the storage indication working curve method. An example
problem utilizing the method is presented later in this chapter.
124
(15)