Environmental Engineering Reference
In-Depth Information
X
X
V
i
@
c
i
@t
¼
Q
j
c
j
Q
k
c
k
k
a;i
A
i
c
i
þ k
d;i
A
i
c
s;i
þ q
i
j
inflow
k
outflow
@
c
s
;
i
@t
¼ k
a;i
c
i
k
d;i
c
s;i
where the index
i
is used to indicate the
ith
room. Inflow and outflow terms
are extended in order to consider that several other rooms may contribute to
the total inflow and outflow. Sorption coefficients are assumed to be room-
specific, as the involved surfaces may be of different kind. Of course it would
also be possible to consider different types of surfaces with respect to sorption
in each room. With all these extensions the presented approach still leads to
a linear system connecting compartments, which can be solved by the
methods described in the second sub-chapter.
Some caution concerning the applicability of the approach should be
mentioned. Bouhamra and Elkilani (
1999
) use the model to determine sorp-
tion coefficients within an experimental test chamber with controlled in- and
outflow and an installed toluol source. For real apartments or buildings, the
number of uncontrolled parameters increases quite fast and surely limits the
model's applicability for predictive purposes. However, the presented
approach can be useful in hypothetical studies exploring the relevance and
interaction of processes. In any case, the compartment approach is more
justified for gaseous environments, where mixing occurs fast in comparison
to an aquatic environment.
@
@t
c
¼
Bc
þ
f
i
E
o
c
(18.6)
where
c
denotes the vector of the state variables. The number of elements of
c
corresponds thus with the number of compartments. In the matrix
B
the exchange
coefficients are gathered.
E
0
is a diagonal matrix, representing outflow into the
exterior, which is also proportional to the state variable.
f
i
is a source/sink vector,
defining a constant sink or source for each compartment which is independent of
any state variable. For a compartment source the sign of the corresponding element
is positive, for a sink it is negative.
For the following we define:
C
¼
B
E
o
(18.7)
For non-constant input vector
f
i
the general solution is given as:
ð
t
CtÞc
c
ðtÞ¼
exp
ð
þ
exp
ð
CtÞ
exp
ð
CsÞ
f
i
ðsÞds
(18.8)
0
