Abstract
A basic set of equations describing the flows of volume (
J
v
) and solute (
J
s
) across a leaky porous membrane, coupled to the differences of osmotic and hydrostatic pressures d
π
and d
P
has been derived by using general frictional theory. Denoting the mean pore concentration of solute by
c
*
s
and the hydraulic and diffusive conductances by
L
p
and
P
s
/
RT
the equations take the form
J
v
=
L
p
d
P
+ σ
s
L
p
d
π
J
s
=
c
*
s
(1 – σ
f
)
J
v
+
P
s
d
π
/
RT
σ
s
=
θ
(1 –
D
s
V
s
/
D
w
V
w
–
D
s
/
D
o
s
σ
f
= 1 –
θ
D
s
V
s
/
D
w
V
w
–
D
s
/
D
o
s
in which
D
w
and
D
s
are the diffusion coefficients for water and solute in the pore and D
o
s
that for free solution. The relation between the reflection coefficients
σ
s
and
σ
f
for osmosis and ultrafiltration is then given by (σ
s
= σ
f
– (1 – θ) (1 –
D
s
/
D
o
s
), where
θ
is the diffusive-driven: pressure-driven flow ratio. These equations follow from the fact that in leaky pores osmosis occurs by diffusion alone and that there cannot be any Onsager symmetry leading to
σ
s
=
σ
f
. Symmetry holds in the limits where either the pore is small, when
σ
s
=
σ
f
= 1, or where the pore is large when
σ
s
=
σ
f
= 0. Symbols used in the text
c
*
s
,
c
*
w
arithmetic mean solute and water concentration across a pore
c
s
,
c
w
solute and water concentration within a pore
D
o
s
diffusion coefficient in free solution
D
s
diffusion coefficient within the pore core
f
ij
partial molar frictional coefficient between the components
i
and
j
f
sw
,
f
sm
molar frictional coefficients between solute and water, solute and membrane
f
d
wm
molar water–membrane frictional coefficient during diffusive flow
f
p
wm
molar water–membrane frictional coefficient during pressure-driven flow
f
π
wm
molar water–membrane frictional coefficient during osmotic flow
J
d
exchange flow between solute and water
J
v
volume flow
K
i
partition coefficient,
c
i
/
c
*
i
L
d
,
L
pd
osmotic conductivity for volume and exchange flows
L
p
,
L
dp
hydraulic conductivity for volume and exchange flows
P
hydrostatic pressure
P
s
,
P
w
solute and water permeability
r
s
,
r
p
radius of solute molecule and pore
S
(
r
i
) restricted diffusion series for species of radius
r
i
in a pore
V
i
partial molar volume of component
i
v
i
velocity of component
i
β
ratio of the two frictional coefficients
f
p
wm
:
f
π
wm
δ
x
pore length
ɳ
viscosity of water
θ
diffusive-driven: pressure-driven flow ratio
μ
i
chemical potential of the component
i
π
osmotic pressure σ
s
, σ
f
osmotic and hydraulic reflection coefficients
ϕ
Rayleigh dissipation function
Reference20 articles.
1. Anderson J. M. & Malone D. M. 1974 Mechanism of osmotic flow in porous membranes. Biophys. J. 14 957-982.
2. Solvent flow in osmosis and hydraulics: network thermodynamics and representation by bond graphs;Atlan H.;Am. J. Physiol.,1987
3. Irreversible thermodynamics and frictional models of membrane processes, with particular reference to the cell membrane. J. theor;Dainty J.;Biol.,1963
4. The frictional coefficients of the flows of non-electrolytes through artificial membranes. J. gen;Ginzburg B. Z.;Physiol.,1963
5. Osmosis
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