Biomedical Engineering Reference
In-Depth Information
−
dQ
−
dQ
⎛
⎞
(
)
(10.35)
∑
B
=
D
=
LP
Δ
−
T
σ
C C
−
⎜
⎟
p
i
B i
,
D i
,
⎝
⎠
dA
dA
i
σ
i
represents the Staverman reflection
coefficient for solute
i
. For a single solute of urea, (10.35) reduces to
where
L
p
is the hydraulic conductance and
dQ
dQ
(
)
(10.36)
−
B
= −
D
=
LP
⎡
Δ
−
TC C
σ
−
⎤
=
K
′
⎣
⎦
p
C
BU
DU
dA
dA
Substituting (10.36) into (10.33) and rearranging
dC
[
]
(10.37)
Q
BU
=−
K
′
K
C
+
K C
B
U
BU
U
DU
dA
Similarly,
dC
(
)
(10.38)
Q
DU
=
K
C
−
C
−
K C
′
D
U
BU
DU
DU
dA
Based on the relative flow direction of each of the two fluids, blood and dia-
lysate, two scenarios are possible: flow may be cocurrent (same direction) or coun-
tercurrent (opposite direction). For analysis involving cocurrent dialysate flow,
(10.36) and (10.38) have opposite signs. One could solve (10.32) and (10.36)
through (10.38) using numerical integration technique to obtain the loss of urea
and water as a function of time. The simplest solution for predicting urea loss as a
function of time is when the assumptions of constant flow rates (
K
′
≈
0 and/or
K
′
<<
K
C
) are valid, such that
Q
B,out
=
Q
B,in
. Integrating (10.33) and (10.34) results in
C
=
C
(1
−
E
)
+
C
E
(10.39)
BU out
,
BU in
,
DU in
,
where
E
is the extraction ratio given by
(
)
1exp
−⎡−
T
Nz
1
+⎤
⎣
⎦
E
=
cocurrent flow
(10.40)
(
)
1
+
z
(
)
exp
⎡
Nz
1
−
⎤ −
1
countercurrent flow
⎣
⎦
T
E
=
(10.41)
(
)
exp
⎡
Nz
1
−
⎤ −
z
⎣
⎦
T
