Passive Mass Transport Processes in Cellular Membranes and their Biophysical Implications - Porous Media: Applications in Biological Systems and Biotechnology - page 310

When both ∆ P and ∆Π exist on the membrane, then the total solute flux
j sM = ∆ m

A ∆ t across the membrane is the sum of the diffusive and convective flux

j sM = ω d ∆Π+(1 − σ ) c s L p ∆ P (8.59)

Equations (8.43) and (8.59) are the mechanistic (ME) transport equations.
The coecient ω d given by equation (8.58) also satisfies:
ω d = j sd
∆Π

(8.60)

∆ P =0

Equation (8.59) for the solute flux can also be expressed as the solute volume
flux

J vsM = ω d V s ∆Π+(1 − σ ) c s V s L p ∆ P (8.61)

As we can notice in equations (8.48 and 8.49) the total volume flux

J vw

J vs
J vsd + J vsk = J vw + J vs J vwa + J ∆ P

vwb + J ∆Π

J vM = J va + J vb =

vwb +

(8.62)

J va J vb

is the sum of J vwa , J ∆ P

vsb , J ∆Π vsb , J vsd , and J vsk .

8.4.3 Correlation Relation for Parameters L p , σ , and ω d
It is known that the three phenomenological transport parameters are not
independent. For instance, this allowed elimination of the reflection coecient
in the 2P formalism. Using the ME described in the previous section we can
derive a correlation relation for the transport parameters in a form:

ω d =(1 − σ ) c s L p (8.63)

To that end we consider equation (8.52), where the fluxes J vM and J vs
are given by equations (8.43) and (8.61). In a case when ∆ P =

− ∆Π these

equations assume forms

J vM = − L p ∆Π − L p σ ∆Π (8.64)

J vs = ω d V s ∆Π − (1 − σ ) c s V s L p ∆Π (8.65)

The water volume flux J vwa is

J vw = J vwa + J ∆ P

vwb + J ∆Π

(8.66)

vwb

where, according to equations (8.39) and (8.45), J vwa equals

J vwa = J va = L p σ ∆ PL p σ ∆Π (8.67)