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.
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