RIVERS AND STREAMS - Water Quality Engineering in Natural Systems - page 89

4.3 MODELS OF SPILLS and setting the partial derivative of the result with

respect to t equal to zero. In the case of a conservative
contaminant ( k = 0), the time, t 0 , of peak concentration
at location x is given by (Hunt, 1999)

Spills of contaminants in rivers are typically associated
with major accidents on transportation routes across or
adjacent to rivers, although other mechanisms, such as
illicit dumping and spikes in continuous wastewater dis-
charges, also occur. Spills are characterized by the intro-
duction of a large mass of contaminant in a very short
period of time.

2

Vt
x

K
Vx

K
Vx

.

(4.38)

0 L L

= − + + 1

This relationship includes the Péclet number, Pe, where

4.3.1 Substances with First-Order Decay

= Vx
K

Pe (4.39)

The governing equation for the longitudinal dispersion
of contaminants that are well mixed over the cross sec-
tions of streams and undergo first-order decay is given
by Equation (4.29). Analytic solutions to this equation
depend on the contaminant release conditions at the
spill location. The most common spill conditions are
instantaneous releases and continuous releases for a
finite time interval.

L

which measures the relative magnitudes of the advec-
tive and diffusive transport. In terms of Pe, Equation
(4.38) can be expressed as

Vt
x

1 1

(4.40)

0

= − + + 1

Pe Pe

2

4.3.1.1 Instantaneous Release. The solution of Equa-
tion (4.29) for the case in which a mass, M , of contami-
nant is instantaneously mixed over the cross section of
the stream at time t = 0 is given by

In most cases, Pe >> 1 and Equation (4.40) yields

x
V

(4.41)

t

0 =

− kt 2

Me ( x Vt

K t

− )

which indicates that the maximum concentration at a
distance x downstream of an instantaneous spill will
occur at a time equal to the mean travel time from the
spill location.

c x t ( , ) = exp −

(4.37)

4

A 4 π K t L

L

where c is the contaminant concentration in the stream
(ML −3 ), x is the distance downstream of the spill (L), t
is the time since the spill (T), A is the cross-sectional
area of the stream (L 2 ), V is the average velocity in the
stream (LT −1 ), and k is the first-order decay factor (T −1 ).
The concentration distribution described by Equation
(4.37) is illustrated in Figure 4.3 for the case of k = 0.
In many cases, we are concerned with the maximum
concentration that will occur at a distance x downstream
of the spill. This maximum concentration can be derived
by taking the logarithm of both sides of Equation (4.37)

EXAMPLE 4.7

Ten kilograms of a conservative contaminant ( k = 0) are
spilled in a stream that is 15 m wide, 3 m deep (on
average), and has an average velocity of 35 cm/s. The
contaminant is rapidly mixed over the cross section of
the stream. (a) Derive an expression for the contami-
nant concentration as a function of time 500 m down-
stream of the spill. (b) If a peak concentration of 4 mg/L
is observed 500 m downstream of the spill, estimate the
longitudinal dispersion coefficient in the stream. (c)
Using the result in Part (b), what would be the maximum
contaminant concentration 1 km downstream of the
spill? (d) If the detection limit of the contaminant is
1 µ g/L, how long after release will the contaminant be
detected 1 km downstream of the release point?

Solution

Vt x

(a) Since the width of the stream, w , is 15 m, and the
average depth, d , is 3 m, the cross-sectional area, A ,
of the stream is given by

Figure 4.3. Concentration distribution resulting from an
instantaneous spill.