Biomedical Engineering Reference
In-Depth Information
V
1
q
1
c
1
V
2
q
2
c
2
Δ
x
FIGURE 7.4
Two-compartment model with a membrane of width
D
x
¼
dx
.
Consider the system of two compartments shown in Figure 7.4, where
V
1
and
V
2
are the volumes of compartments 1 and 2
q
1
and
q
2
are the quantities of solute in compartments 1 and 2
c
2
are the concentrations of solute in compartments 1 and 2
and an initial amount of solute,
c
1
and
Q
10
, is dumped into compartment 1. After approximating
c
the derivative
dc
dx
as
c
1
2
, the rate of change of solute in compartment 1 is given by
D
x
ð
c
Þ
q
1
¼
DA
dc
dx
¼
DA
c
1
2
ð
7
:
2
Þ
D
x
Next, the quantity is converted into a concentration by
q
¼
V
ð
7
:
3
Þ
c
1
1
1
and after differentiating Eq. (7.3), gives
q
1
¼
V
c
ð
7
:
4
Þ
1
1
Substituting Eq. (7.4) into Eq. (7.2) yields
V
1
c
1
¼
DA
D
x
ð
c
1
c
2
Þ
ð
7
:
5
Þ
With the transfer rate
K
defined as
K
¼
DA
D
x
when substituted into Eq. (7.5) yields
c
1
¼
K
V
1
c
1
c
2
ð
Þ
ð
7
:
6
Þ
From conservation of mass, we have
Q
¼
q
þ
q
10
1
2
which after converting to a concentration gives
V
C
¼
V
c
þ
V
c
ð
7
:
7
Þ
1
10
1
1
2
2
C
10
¼
Q
10
V
1
where
is the initial concentration in compartment 1 due to the initial amount of
solute dumped into the compartment.