Environmental Engineering Reference
In-Depth Information
where indices 1-3 refer to soil, roots and leaves, respectively; C (mg kg 1 ) is con-
centration; k 1 , k 2 and k 3 are the sum of all first-order loss processes in compartment
1, 2 and 3, respectively, and k 12 and k 23 are the transfer rates from compartment
1 to 2 and 2 to 3, respectively. I i (mg d 1 ) describes the constant input to the
compartments, e.g., from air, and M i (kg) is the mass of compartment i , i
1, 2,
3. The matrix elements k and I can be derived from the differential equations above
(Eqs. 9.15 , 9.18 and 9.20 ).
Linear differential equations approach steady state for t
=
→∞
, i.e. the change
of concentration with time is zero, dC/dt
0. The steady-state solutions for matrix
equations 1 (soil), 2 (roots) and 3 (leaves) with continuous input are as follows:
=
I 1
k 1 M 1
C 1 ( t
→∞
)
=
(9.29)
I 2
k 2 M 2 +
k 12
k 2
C 2 ( t
→∞
)
=
C 1 ( t
→∞
)
(9.30)
I 3
k 3 M 3 +
k 23
k 3
C 3 ( t
→∞
)
=
C 2 ( t
→∞
)
(9.31)
The steady-state solution follows the general scheme:
I n
k n M n +
k n 1, n
k n
C n ( t
→∞
)
=
×
C n 1 ( t
→∞
)
(9.32)
where n is the compartment number.
The analytical solutions for the differential equations 1 (soil), 2 (roots) and
3 (leaves) for a pulse input is the same as for initial concentrations C(0)
=
0:
e k 1 t
C 1 ( t )
=
C 1 (0)
×
(9.33)
e k 1 t
( k 2
e k 2 t
( k 1
e k 2 t
C 2 ( t )
=
k 12 C 1 (0)
×
k 1 ) +
+
C 2 (0)
×
(9.34)
k 2 )
e k 1 t
e k 2 t
e k 3 t
C 3 ( t ) = k 12 k 23 C 1 (0) ×
k 3 ) +
k 3 ) +
( k 1
k 2 )( k 1
( k 2
k 1 )( k 2
( k 3
k 1 )( k 3
k 2 )
(9.35)
e k 2 t
( k 3
e k 3 t
( k 2
+ C 3 (0) × e k 3 t
k 2 ) +
+ k 23 C 2 (0) ×
k 3 )
The general solution scheme for pulse input to soil only, i.e. C 1 (0)
=
0 and C n (0)
=
0 with n > 2 is as follows:
n
1
n
e k j t
C n ( t )
=
k i , i + 1 C 1 (0)
×
(9.36)
n
( k k
k j )
i
1
j
=
1
k = 1, k = j
 
Search WWH ::




Custom Search