Biomedical Engineering Reference
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that its definition is different from Staverman's. A definition of the reflection
coecient of the entire membrane will be given later but if a given membrane
has only impermeable pores we treat it as a semipermeable membrane and
assign it the reflection coecient σ =0 . If there are only permeable pores
then we say the entire membrane is permeable and has σ =1 . In case of a
membrane with both permeable and impermeable pores we divide the mem-
brane into two parts: (a) and (b). For a given solute, all impermeable pores
are in part (a) of the membrane and all permeable pores are in part (b). We
can also allow a number of unstable pores that will change their dimensions
and change from being impermeable to permeable and vice versa. We assume,
however, that statistically the numbers of permeable and impermeable are
constant.
8.3.2 Poiseuille's Equation for Individual Pores and
for the Membrane
Detailed models of water permeation in pores, especially biological, have been
dicult to develop. When the pores are very narrow, with sizes comparable
to the dimensions of water molecules, a molecular description is required, but
is not available. For larger pores, standard techniques of hydrodynamics can
be used. For the lack of better description, here we are using hydrodynamical
concepts to describe permeation through individual pores; however, it needs
to be understood that the validity of the approximation becomes questionable
as pores get more and more narrow (Figure 8.2).
Since membrane pores have hydrophilic walls and are filled with water,
motion of water molecules in the pores is determined by cohesion and adhesion
forces. The narrowest pores are the least permeable, and, as the pore diameter
increases, the molecules move with increasing ease. We assume the Poiseuille's
v
P 2 >
A
P 1
x
FIGURE 8.2
A cylindrical pore ( A is the cross-section; P 1 ,P 2 is the pressures on both sides
of the membrane; ∆ x is the pore length; v is the average flow speed in the
pore).
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