Biomedical Engineering Reference
In-Depth Information
of carbon-carbon chain between two carbonyl groups/a carbonyl group and an
adjacent hydroxymethyne group, which produces smaller molecules of random
sizes.
In order to mathematically model endogenous depolymerization processes of
polymers such as PVA, let
wtM
(, ) be its weight distribution with respect to the
molecular weight
M
at time
t
. Denote by
CAB
(,) the class of all molecules whose
molecular sizes lie between
A
and
B
. Then the total weight of
CAB
(,) present at
time
t
is the integral of
wtM
(,
) with respect to
M
over the interval [,]
AB
B
∫
(12.21)
wtM M
(,
)d
A
For
KM
(, )
due to endogenous depolymerization. The transition of the weight from
CAB
≤
, let
ptK M
(, ,
) denote the time rate of transition from
wtM
(,
) to
wtK
(,) to
CDE
(,) per unit time is the integral of
ptK M
(, ,
) with respect to (, )
KM
over the
region
R
∫∫
ptK M M K
(, ,
)d
d
R
where
RKMKMDKEAMB
=
{(
,
) |
≤
,
≤
≤
,
≤
≤
}
The total weight decrease in
CAB
(,) per unit time is given by
B
M
∫
∫
ptK M K M
(, ,
)d
d
(12.22)
A
0
while the total weight increase per unit time is given by
B
∞
∫
∫
ptMK
(,
, )d
K M
d
(12.23)
A
M
The equation
d
d
∂
∂
w
t
B
B
∫
∫
wtM M
(,
)
d
=
(,
tM M
)
d
t
A
A
holds for the rate of change of the quantity (12.21), and it equals the difference
between the quantities (12.22) and (12.23)
∂
∂
w
t
B
B
M
B
∞
∫
∫
∫
∫
∫
(,
tM M
)
d
=−
ptKM K M
(, ,
)
d
d
+
ptMK K M
(,
, )
d
d
A
A
0
A
M
which leads to
{
}
∂
∂
w
t
B
M
∞
∫
∫
∫
(,
tM
)
+
ptKM K
(, ,
)
d
−
ptMK K
(,
, )
d
d
M
=
0
