McKay, Donald. "Front matter" Multimedia Environmental Models ...

y Mackay et al. (1983), and an application to surfactant decay in a river has been
described by Holysh et al. (1985).
A differential equation is set up for the water column as a function of flow
distance or time, and steady-state exchange with the sediment is included. The
equation can then be solved from an initial condition with zero or constant inputs
of chemical. The practical difficulty is that changes in flow volume, velocity, or river
width or depth cannot be easily included, therefore the equation necessarily applies
to idealized conditions.
This equation may be useful for calculating a half-time or half-distance of a
substance in a river as the concentration decays as a result of volatilization or
degradation. A version is the oxygen sag equation. This contains an additional term
for oxygen consumption by organic matter added to the river. This model was first
developed by Streeter and Phelps in 1925 and is described in texts such as that of
Thibodeaux (1996). This is historically significant as being among the first successful
applications of mathematical models to the fate of a chemical (oxygen) in the aquatic
environment.
©2001 CRC Press LLC
8.8 QWASI MULTI-SEGMENT MODELS
A lake, river, or estuary rarely can be treated as a single well mixed “box” of
water, and a more accurate simulation is obtained if the system is divided into a
series of connected boxes. In the case of a river, it may be acceptable to use the
output from one box as input to the next downstream box and treat any downstreamupstream
flow as being negligible compared to upstream-downstream flow. In slowly
moving water, this will be invalid if there are significant flows in both directions.
In principle, if there are n water and n sediment boxes, there are 2n mass balance
equations containing 2n fugacities, and the equations can be solved algebraically
for steady-state conditions or numerically for dynamic conditions.
In the case of steady-state conditions, a major simplification is possible if it is
assumed that there is no direct sediment-sediment transfer between reaches; i.e., all
transfer is via the water column. Each sediment mass balance equation then can be
written to express the sediment fugacity as a function of the fugacity in the overlying
water. The fugacity in sediment then can be eliminated entirely, leaving only n water
fugacities to be solved. This is equivalent to calculating the water fugacity as in the
QWASI model by including loss to the sediment in the denominator. A total loss D
value can thus be calculated for the water and sediment.
Algebraic solution of the equations is straightforward, provided the number of
boxes is small and there is minimal branching.
For a set of boxes connected in series with both “upstream and downstream”
flows, the solution becomes simple, elegant, general, and intuitively satisfying
because of its transparency. This is illustrated in Figure 8.8. For a given box, the
numerator consists of a series of terms, each reflecting inputs (designated I) to this
box and Q, including input from other boxes. For each box, the input I (by discharge
and from the atmosphere) is included directly. For adjacent boxes, the input to that
box is multiplied by the fraction that migrates to the box in question. This fraction,