Passive Mass Transport Processes in Cellular Membranes and their Biophysical Implications - Porous Media: Applications in Biological Systems and Biotechnology - page 309

8.4.2.1 Case 1

Assume that ∆Π = 0, ∆ P> 0 , and the solut i ons on both sides of the mem-
brane are the same and equal c s . Then also c s = c s . The hydraulic pressure
generates a flow of solvent and by convection also of the solute. The convective
solute flux can be written as

j sk = c s J vb

(8.50)

where J vb is given by equation (8.40). Hence

j sk =(1 − σ ) c s L p ∆ P (8.51)

8.4.2.2 Case 2

Assume ∆ P = 0 and ∆Π > 0. Then from (8.47-8.49) we have:

J vb = J ∆Π

vwb + J ∆Π vsb = 0

(8.52)

since σ b =0 ( σ b is the filtration coecient of part [b] of the membrane, see
Figure 8.3). The fluxes J ∆Π
vwb

and J ∆Π
vsb

have opposite directions, as shown

in Figure 8.3. We denote them as J ∆Π
vsb

= J vsd , and J ∆Π

vwb = J vwd , where the
latter is the diffusive volume flux of water relative to the membrane. To show
that J vsd is the diffusive volume flu x of s o lve nt , also measured r elati v e to th e
membrane, we notice that J vsd = j s V s = c s ν s V s and J vwd = j w V w = c w ν w V w .
Since

J vsd = − J vwd (8.53)

we get

c w ν w V w (8.54)
where v s , v w are the velocit ie s of molecules of the solute and water, respec-
tively. For diluted solutions c s

c s ν s V s = −

ν w .In
other words, the velocity of the water relative to the membrane is negligibly
small and the convection of solute by moving water can be disregarded.
The fluxes J vwd and J vsd can be written as

c w ≈ 1; hence we can conclude ν s

J vwd = L pbw c w V w ∆Π (8.55)

J vsd = L pbs c s V s ∆Π (8.56)

where L pbw and L pbs are the hydraulic conductivities for the diffusive perme-
ation of water and the solute through part (b) of the membrane. The latter
formula is also written as

j sd = L pbs c s ∆Π = ω d ∆Π (8.57)

where

ω d = L pbs c s (8.58)

is called the coecient of diffusive permeation of solute.