ARUP; ISBN: 978-0-9562121-5-3 - CMBBE 2012 - Cardiff University

4
R ( dP
/ dz)
/( 8)
ql 1 l
Here, ql varies with z, as fluid leaks out of the lumen, and eventually back into it. This can
dPl 8(
q0
qc
)
4
be written as dz R1
In the cells region (compartment B) the flow is modeled as a percolating flow through the
cells, described by Darcy’s differential equation
k ( dP
/ dz)
q
/
c
this can be written as
dP
c
dz
c
q
c
2
3
k
( R R )
c
c
2
2
c
2 2 ( R R )
3
2
There is a pressure difference across the membrane, and this drives the plasma to percolate
through the region from R1 to R2 and ultimately back again. We model this percolation as a
Darcy flow with Darcy constant m k
.
As the leakage into the cells region in which increases qc we have dqc=2πR2dzυR2, and so
dqc c
dz
2 R1
2k
m ( Pl
P )
ln( R / )
Three simultaneous linear differential equation were derived for Pl , Pc and qc. In principle
three initial conditions are needed to go with them; that is values of Pl, Pc and qc at z=0.
Therefore,
( a)
P P
( b)
q
( c)
q
l
c
c
0,
q
a given valu e, at
0
0,
at z L.
z
0,
q , given valu es, small, at z 0,
4.2 OXYGEN CONSUMPTION RATE DIFFRENTIAL EQUATIONS
In the case of qc increasing with z, the oxygen transfer rate balance for a section dz of the
annulus:
pq
p
dq (
p p)
dz (
p dp)(
q dq ) pdz/(
p K).
c
0
c
Cancelling , neglecting dpdq
G
and rearranging
gives,
dp ( p0
p)
dqc
/ dz p
/( p K)
(
pG
p)
f ( z,
p)
dz
q
c
c
In the case of qc decreasing with z, the balance
( p p)
dz (
p dp)(
q dq ) p(
dq
) pdz
/( p K)
.
pqc G
c c
c
Cancelling and rearranging gives
dp p
/( p K)
(
p
dz
q
c
G
c
p)
g(
z,
p)
c
From an initial condition for p at z=0, a numerical procedure is used to obtain the solution
in steps using dp/ dz f ( z,
p)
until dqc/dz changes sign from positive to negative. From the