Page 74 - Basic physical chemistry for the atmospheric sciences

60 Basic physical chemistry

displaces the water lying above it, by pushing it upward without any
mixing, the time that each small element of water that enters the
bottom of the tanks spends in the tank, before overflowing at the top,
is MIF, where M is the volume of the tank [hence the reason for
)
defining residence time as MIF in Eq . (3. 1 1 ) . In this case, when no
mixing occurs in the reservoir, the residence time of the water is the
same as the renewal time ( T) , which is defined as the time required to
completely displace the original water from the tank.
Now let us consider a more realistic situation for natural system ;
s
i
y
namel , one n which mixing takes place between the material that is
injected into the reservoir and the material that is already residing in
the reservoir. For simplicity, we will consider the mixing to be com­
plete and thorough (i. e . , perfect mixing). The tank representation of
our reservoir is again helpful. Suppose that at time zero the tank is full
of dirty water, and at this time clean water starts to be pumped into
the bottom of the tank. Since the mixing is perfect, the rate of removal
of dirty water from the top of the tank will be proportional to the
fraction of the water in that tank that is dirty water. Therefore, if W is
the amount of dirty water in the tank at time t
dW
- = - k W (3 . 1 3 )
dt

where k is a constant of proportionality. From Eqs. (3 . 1 1 ) and (3. 1 3)
we have for the dirty water
W 1
T = ( - dWfd t ) - k (3. 1 4)

Since Eq . (3. 1 3) has the same form as Eq. (3.4), the half-life of the
dirty water is given by Eq. (3. 1 0 ) . Combining Eqs. (3. 1 0) and (3. 1 4 ,
)
we obtain the following relationship between the half-life (tl / ) and the
2
residence time ( ) for the case of perfect mixing
T
t 1/2 = Q.693T (3 1 5 )
.
(
In the case of perfect mixing, the renewal time T) will be infinitely
long, since some molecules of dirty water will always be present in the
tank. However, we can obtain an approximate relationship between T,
t 1/2 and T for perfect mixing as follows. From the definition of the half­
life (t 12 ) , we know that after t 112 minutes one-half of the dirty water
1
will be left in the tank, and after 2t z minutes ( 1 / 2)(1 / 2) = ( 1 / 2) of the
2
l l