Passive Mass Transport Processes in Cellular Membranes and their Biophysical Implications - Porous Media: Applications in Biological Systems and Biotechnology - page 313

The ME solute flux equation

j sM = ω d ∆Π+(1 − σ ) c s L p ∆ P

and the KK equation:

j s = ω ∆Π + c s (1 − σ ) J v

have different forms. The former was written in the following 2P form:

j sM =(1

− σ ) c s L p (∆ P +∆Π)=(1 − σ ) c s L p ∆Π+(1 − σ ) c s L p ∆ P (8.82)

where (1 σ ) c s L p ∆ P=j sk is

the convection flux of the solute. To bring the KK to the analogous 2P form,
we derive first a correlation relation for the KK transport parameters L p , σ,
and ω . Let us recall that the total volume flux is given by

− σ ) c s L p ∆Π =j sd is the diffusion flux and (1 −

J v = j w V w + j s V s = J vw + j s V s

Considering equations (8.59) and (8.60) we obtain

J v = j vw + ωV s ∆Π+(1 − σ ) c s V s ( L p ∆ P − L p σ ∆Π) (8.83)

For dilute solutions ( c w c s ) we can assume J v ≈ J vw , hence

0= ω ∆Π+(1 − σ ) c s L p ∆ P − (1 − σ ) c s L p σ ∆Π (8.84)

Assuming ∆ P = − ∆Π we get the correlation equation

σ 2 ) c s L p

ω =(1 − (8.85)

Substituting in the KK equation we get its 2Ps form

j s =(1 − σ ) c s L p (∆ P +∆Π)=(1 − σ ) c s L p ∆Π+(1 − σ ) c s L p ∆ P (8.86)

which is identical with the reduced ME equation for the solute flux. Also, con-
sidering correlation formulas (8.59), (8.85) we can find the connection between
the transport parameters in the form:

ω =(1+ σ )(1 − σ ) c s L p = ω d (1 + σ ) (8.87)