Environmental Engineering Reference
In-Depth Information
The entropy in the
u
-th order river is
d
H
d
Y
<
³
u
k
³
u
(1.7)
u
Y
Y
u
u
where
H
u
= average potential energy in the
u
-th order river;
Y
u
= average total fall or change of elevation
in the
u
-th order river; and
k
= a conversion factor between potential energy and elevation. The distribution
of potential energy per unit weight of water in a river system is directly proportional to its elevation. The
probability that a particular amount of potential energy will be lost in the
u
-th order river is
H
Y
p
u
u
(1.8)
u
H
Z
m
Substituting Eq. (1.8) into Eq. (1.7)
d
p
³
<
k
u
k
ln
p
c
(1.9)
u
u
u
P
u
In which
c
u
is a constant. The total entropy of a system is equal to the sum of the entropy of each part.
m
m
¦¦
<
<
k
ln
p
c
(1.10)
u
u
u
1
u
1
In which
c=c
1
+c
2
+
Ă
+c
m .
According to Lewis and Randall (1961), the most likely distribution of energy
in a system under dynamic equilibrium condition is that the entropy of the whole system is a maximum, i.e.,
m
maximum
(1.11)
ln
p
=
u
u
1
m
¦
From the definition of the probability,
p
1
. The entropy reaches its maximum value as the water
u
u
1
head evenly distributed, i.e.,
"
(1.12)
According to Prigogine (1967), during the evolution toward a stationary state, the rate of entropy
production per unit mass or weight should be a minimum compatible with external constraints.
d
d
pp
p
1
2
m
Y
t
minimum
which can be expressed as
ddd
ddd
YYx
sV
minimum
(1.13)
t
x
t
In which
x
is the distance along the flow direction,
s
is energy slope,
V
is velocity. Integration of Eq. (1.13)
yields
³
sVA Qs
d
minimum
(1.14)
In which d
A
is the differential of wet area,
Q
is the discharge, and s is riverbed slope, which is equal to
the energy slope in steady flows.
Define a stream power
P s
J
,
Eq. (1.14) can be rewritten as:
P s
J
minimum
(1.15)
In which J is the specific weight of water.
Equation (1.15) states Yang's law of least rate of energy expenditure that during the evolution toward
its equilibrium condition, a river will adjust or choose its course of flow in such a manner that the rate of
potential energy expenditure per unit mass or weight of water is a minimum. The minimum value depends
on the constraints applied to the system (Yang, 1972).
Search WWH ::
Custom Search