RIVERS AND STREAMS - Water Quality Engineering in Natural Systems - page 83

where m is the mass of substance (kg), c p is the specific
heat at constant pressure (J/kg·K), and T is the tempera-
ture (K). The specific heat, c p , of natural waters varies
only slightly with temperature, and for natural waters, a
value of 4190 J/kg·K is a satisfactory approximation.
Equating the heat content of the mixed water to the
sum of the heat contents of the river and waste dis-
charge, and assuming that the density and specific heat
remains constant, gives

completely mixed plume

river bank

river

B

A

Q r + Q w , c 0

Q r , c r

Q w , c w

Q c T Q c T ρ + ρ = ( Q Q + ) ρ c T (4.25)

waste discharge

partially mixed plume

r r p r w w p w w r m p m

Figure 4.2. Initial mixing of stream discharges.

where ρ r , ρ w , and ρ m are the densities of the river, waste
discharge, and mixed waters, respectively (assumed to
all be equal), and T r , T w , and T m are the temperatures of
the river, waste discharge, and mixed waters, respec-
tively. Simplifying Equation (4.25) yields

(L 3 T −1 ), with a contaminant concentration c w (ML −3 ).
The wastewater discharge occurs at section A, the efflu-
ent is completely mixed across the river at section B,
and conservation of contaminant mass requires that

QT Q T
Q Q

+
+

r r w w

T

0 =

(4.26)

Q c Q c + = ( Q Q c + )

0

(4.21)

r r w w r w

r w

where c 0 is the concentration of the wastewater/river
mixture at the section where the effluent is completely
mixed across the cross section (i.e., section B). Rear-
ranging Equation (4.21) gives the following expression
for the concentration of the diluted wastewater imme-
diately after it is completely mixed with the ambient
river water:

This equation is commonly used to determine the tem-
perature of water mixtures where the densities and spe-
cific heats of the individual components of the mixture
are approximately equal to the density and specific heat
of the mixture.
In models that use the one-dimensional (along-
stream) advection-diffusion equation (ADE) to
describe the mixing of contaminants in streams, an
initial concentration c 0 (ML −3 ) is usually assumed to
occur at the wastewater discharge location. It is clear
from Equation (4.22) that for a given wastewater dis-
charge, lower stream flows will result in higher concen-
trations of the diluted wastewater. In analyzing the fate
and transport of municipal and industrial waste dis-
charges into streams, the 7-day average low low with a
return period of 10 years is usually used as the design
flow in the stream. Such flows are typically written using
the notation aQb , where a is the number of days used
in the average, and b is the return period in years of
the minimum a -day average low low. Therefore, 7 Q 10
is the 7-day average low low with a return period of 10
years.

Q c Q c
Q Q

+
+

r r w w

c

0 =

(4.22)

r w

This conservation-of-mass-flux approach can also be
used to estimate contaminant concentrations down-
stream of the confluence of two or more streams, where
the concentration, c (ML −3 ) downstream of the conflu-
ence of N streams is given by

N

∑

Qc

i i

i = 1

c = (4.23)

N

∑

Q

i

i = 1

where Q i (L 3 T −1 ) and c i (ML −3 ) are the flow rate and
concentration in stream i , respectively.
In the case of mixing waters having different tem-
peratures, a heat balance rather than a mass balance
must be done. The heat content (also called enthalpy ),
H (J), of a mass of a substance is given by

EXAMPLE 4.4

The Do concentration in a river upstream of a munici-
pal wastewater outfall is 10 mg/L. (a) If the 7 Q 10 river
flow upstream of the outfall is 50 m 3 is and the outfall
discharges 2 m 3 /s of wastewater with a Do concentra-
tion of 1 mg/L, estimate the Do concentration in the
river after complete mixing. (b) If the upstream river H mc T = (4.24)

p